bcpldtpbc 201701 jee 17-18

CONIC SECTION

(PARABOLA, ELLIPSE & HYPERBOLA)

JEE (MAIN+ADVANCED)

CONIC SECTION

(PARABOLA, ELLIPSE & HYPERBOLA)

S.No Pages

1. Theory 01 – 51

2. Exercise-1 (Special DPP) 52 – 72

3. Exercise-2 72 – 75

4. Exercise-3 (Section-A) 76 – 85

[Previous years JEE-Advanced problems]

5. Exercise-3 (Section-B) 85 – 88

[Previous years JEE-Main problems]

6. Exercise-4 (Potential Problems for Board Preparations) 89

7. Exercise-5 (Rank Booster) 90 – 91

8. Answer Key 92 – 95

CONTENT

CONIC SECTION

BANSAL CLASSES Private Ltd. ‘Bansal Tower’, A-10, Road No.-1, I.P.I.A., Kota-05 Page # 1

CONIC SECTION

Point, pair of straight lines, circle, parabola, ellipse and hyperbola are called conic section because they

can be obtained when a cone (or double cone) is cut by a plane.

The mathematicians associated with the studyof conics were Euclid, Aristarchus and Apollonius. Most

of the objects around us and in space have shape of conic-sections. Hence study of these becomes a

very important tool for present knowledge and further exploration.

The conic section is the locus of a point which moves such that the ratio of its distance from a fixed

point (focus) to perpendicular distance from a fixed straight line (directrix) is always constant (e). Here

e is called eccentricity of conic i.e.,

ePM

PS M

directrix

Ax+By+C=0

P(x, y)

S( , )focus axis

A line through focus and perpendicular to directrix is called - axis. The vertex of conic is that point where

the curve intersects its axis.

222 PMePSePM

PS

2

22

222

BA

CByAxe)y()x(

Simplification shall lead to the equation of the form ax2 + by2 + 2hxy + 2gx + 2fy + c = 0

Distinguishing various conics :

The nature of the conic section depends upon the position of the focus S w.r.t. the directrix & also

upon the value of the eccentricity e. Two different cases arise.

Case-I : When The Focus Lies On The Directrix (De-generated conic) :

In this case abc + 2fgh af2 bg2 ch2 = 0 & the general equation of a conic represents a pair

of straight lines if

e > 1 i.e. h2 > ab the lines will be real & distinct, intersecting at S.

e = 1 i.e. h2 = ab the lines will coincident.

e < 1 i.e. h2 < ab the lines will be imaginary.

CONIC SECTION


Case-II : When The Focus Does Not Lie on the Directrix (Non de-generated conic) :

In this case abc + 2fgh – af 2 – bg2 – ch2 0and conic represent

a parabola an ellipse a hyperbola rectangular hyperbola Circle

e = 1 0 < e < 1 e > 1 e = 2 e = 0

h² = ab h² < ab h² > ab h² > ab; a + b = 0 h = 0, a = b

Note :(i) For pair of straight lines e(ii) All second degree terms in parabola form a perfect square

Definition of various terms related to a conic :

(1) Focus : The fixed point is called a focus of the conic.

(2) Directrix : The fixed line is called a directrix of the conic.

(3) Axis : The line passing through the focus and perpendicular to the directrix is called the axis of theconic.

(4) Vertex : The points of intersection of the conic and the axis are called vertices of the conic.

(5) Centre : The point which bisects everychord of the conic passing through it, is called the centre of theconic.

(6) Latus-rectum : The latus-rectum of a conic is the chord passing through the focus and perpendicular tothe axis.

(7) Double ordinate : Achord which is perpendicular to the axis of parabola or parallel to its directrix.

CONIC SECTION


PARABOLA

A parabola is the locus of a point which moves in a plane such that its distance from a fixed point (focus)is equal to its perpendicular distance from a fixed straight line (directrix). Eccentricity of parabola is 1.

Standard Equation of a parabola :

Let S be the focus and ZN is the directrix of the parabola.From S, draw SZ perpendicular to the directrix.Let O be the middle point of ZS. Take O as the origin and OSas x-axis and OY perpendicular to OS as the y-axis.Let ZS = 2a, then ZO = OS = a

M

N

Z

x = –a

O S(a, 0)

P

X

Y

Now, S (a, 0) and the equation of ZN is x = –a or x + a = 0.Let P(x, y) be any point on the parabola. PS = PM (by definition of parabola).

22 )0y()ax( =01

|ax|2

22 y)ax( = |x + a|

or (x – a)2 + y2 = (x + a)2

or x2 – 2ax + a2 + y2 = x2 + 2xa + a2

or y2 = 4ax which is the required equation.

Terms related to Parabola :

y

M L(a,2a)

S (a,0)x

A

Z

Directrix x = a

Focusaxis

Focal chord

Double ordinate

Latus Rectum

Vertex

Focal distance

y’

x + a = 0

P

(a,-2a)L’

tangent at vertex

(1) Axis : Astraight line passes through the focus and perpendicular to the directrix is called the axis ofparabola. For the parabola y2 = 4ax, x-axis is the axis.Since equation has even power of y therefore the parabola is symmetric about x-axis i.e. about its axis.

(2) Vertex : The point of intersection of a parabola and its axis is called the vertex of the Parabola. For theparabola y2 = 4ax, O(0, 0) is the vertex.The vertex is the middle point of the focus and the point of intersection of axis and directrix.

CONIC SECTION


(3) Focal Distance : The distance of any point P (x, y) on the parabola from the focus is called the focallength (distance) of point P.The focal distance of P = the perpendicular distance of the point P from the directrix.

(4) Double Ordinate : The chord which is perpendicular to the axis of Parabola or parallel to Directrix iscalled double ordinate of the Parabola.

(5) Focal Chord : Any chord of the parabola passing through the focus is called Focal chord.

(6) Latus Rectum : If a double ordinate passes through the focus of parabola then it is called as latusrectum. The extremities of the latus rectum are L (a, 2a) and L'(a, –2a). Since LS = L'S = 2a, thereforelength of the latus rectum LL' = 4a.

(7) Parametric Equation of Parabola : The parametric equation of Parabola y2 = 4ax are x = at2, y= 2at.Hence any point on this parabola is (at2, 2 at) which is also called as 't' point.

Note:(i) The length of the latus rectum = 2 × perpendicular distance of focus from the directrix.(ii) If y2 = lx then length of the latus rectum = l .(iii) Two parabolas are said to be equal if they have same latus rectum.(iv) The ends of a double ordinate of a parabola can be taken as (at2, 2 at) and (at2, – 2at).(v) Parabola has no centre, but circle, ellipse, hyperbola have centre.

Other Standard parabola :

Equation of parabola y2 = 4ax y2 = – 4ax x2 = 4ay x2 = – 4ay

(a) Graphs Z O S S O Z

S

O

Z

O

Z

S

(b) Eccentricity e = 1 e = 1 e = 1 e = 1

(c) Focus S(a, 0) S(–a, 0) S(0, a) S(0, –a)(d) Equation of directrix x + a = 0 x – a = 0 y + a = 0 y – a = 0

(e) Equation of axis y = 0 y = 0 x = 0 x = 0(f) Vertex O(0, 0) O(0, 0) O(0, 0) O(0, 0)

(g) Extremitiesof (a, ±2a) (–a, ±2a) (±2a, a) (±2a, –a)latusrectum

(h) Length of latusrectum 4a 4a 4a 4a

(i) Equation of tangent x = 0 x = 0 y = 0 y = 0at vertex

(j) Parametric coordi-nates of any point P(at2, 2at) P(–at2, 2at) P(2at, at2) P(2at, –at2)on parabola

CONIC SECTION


Equation of parabola with respect to two perpendicular lines :

Let P(x, y) is any point on the parabola then equationof parabola y2 = 4ax is consider as

(PM)2 = 4a(PN)x

y

N

M

P(x,y)

Oi.e. (The distance of P from its axis)2 = (latus-rectum)(The distance of P from the tangent at its vertex)where P is any point on the parabola.

CHORD :Line joining any two points on the parabola is called its chords.

Let the points P(at12, 2 at1) and Q(at2

2, 2 at2), lie on the parabola then equation of chord is

(y – 2at1) = 21

22

12

atat

at2at2

(x – at1

2)t1

t2

(c, 0)

t3

t4

Y

XO

y – 2at1 =21 tt

2

(x – at12)

(t1 + t2) y = 2x + 2at1 t2If this chord meet the x-axis at point (c, 0) then from above equation

c + at1t2 = 0 i.e. t1t2 = –c/a.

Note:(i) If the chord joining t1, t2 & t3, t4 pass through a point (c, 0) on the axis, then t1t2 = t3t4 = c/a.

(ii) If PQ is a focal chord then t1t2 = – 1 or t2 = –1t

1. which is required relation.

Hence if one extremity of a focal chord is (at2, 2at) then the other extremity will be

t

a2,

t

a2 .

Position of a point with respect to a parabola y2 = 4ax :

Let P(x1, y1) be a point. From Pdraw PMAX (on the axis of parabola) meeting the parabola y2 = 4axat Q (x1, y2) where Q (x1, y2) lie on the parabola therefore

22y = 4ax1 …(1)

Now, P will be outside, on or inside the parabolay2 = 4ax according as

PM >, = , or < QM

Y

X

P(x ,y )1 1

Q(x ,y )1 2

M

y =4ax2

AX'

Y'

Interiorregion

Exteriorregion

(PM)2 >, = , or < (QM)2

y12 >, = , or < y2

2

y12 >, = , or < 4ax1 (from(1))

Hence y12 – 4ax1 >, =, or < 0

Hence in short, equation of parabola S(x, y) = y2 – 4ax.

(i) If S(x1, y1) > 0 then P(x1, y1) lie outside the parabola.(ii) If S(x1, y1) < 0 then P(x1, y1) lie inside the parabola.(iii) If S(x1, y1) = 0 then P(x1, y1) lie on the parabola.

This result holds true for circle, parabola and ellipse.

CONIC SECTION


Note :(i) The length of focal chord having parameters t1 and t2 for its end points is a(t2 – t1)

2.

(ii) 2t

1t for all t 0 ( AM GM)

a4t

1ta

2

Length of focal chord latus rectumi.e., The length of smallest focal chord of the parabola is 4a, which is the latus rectum ofa parabola.

Note : If l1 and l2 are the length of segments of a focal chord of a parabola, then its latus rectum is21

21

ll

ll4

.

INTERSECTION BETWEEN THE LINE AND PARABOLA :

Let the parabola be y2 = 4ax ... (i)and the given line be y = mx + c ... (ii)then line maycut, touch or does not meet parabola.

The pointsof intersection of the line (1) and theparabola (2) will beobtained bysolving the two equationssimultaneously. Bysolving equation (i) and (ii), we get

my2 – 4ay + 4ac = 0this equation has two roots and its nature will be decided by the discriminant D = 16a(a – cm)

Now, if D > 0 i.e., c <m

a, then line intersect the parabola at two distinct points.

If D = 0 i.e., c =m

a, then line touches the parabola. (It is condition of tangency)

If D < 0 i.e., c >m

a, then line neither touch nor intersect the parabola.

EQUATION OF TANGENT :

1. Point Form :

Equation of parabola is y2 = 4ax .........(1)Let P (x1, y1) and Q = (x2, y2) be any two points on parabola (1), then

y12 = 4ax1 .........(2)

and y22 = 4ax2 ........(3)

Subtracting (2) from (3) theny2

2 – y12 = 4a(x2 – x1)

or1212

12

yy

a4

xx

yy

.......(4)

TP(x , y )1 1

Q(x , y )2 2

Q2

Q1

CONIC SECTION


Equation of PQ is

y – y1 =12

12

xx

yy

(x – x1) .......(5)

From (4) and (5), then

y – y1 =12 yy

a4

(x – x1) ......(6)

Now for tangent at P, Q P, i.e., x2 x1 and y2 y1 then equation (6) becomes

y – y1 =1y2

a4(x – x1)

or yy1 – y12 = 2ax – 2ax1

or yy1 = 2ax + y12 – 2ax1

or yy1 = 2ax + 4ax1 – 2ax1 [From (2)]or yy1= 2ax + 2ax1which is the required equation of tangent at (x1, y1).

The equation of tangent at (x1, y1) can also be obtained by replacing x2 by xx1, y2 byyy1, x by2

xx 1,

y by2

yy1

and xy by

2

yxxy 11 and without changing the constant (if any) in the equation of curve.

This method (standard substitution) is apply to all conic when point lie on the conic.

2. Slope Form :

The equation of tangent to the parabola y2 = 4ax at (x1, y1) isyy1 = 2a (x + x1) ........(1)

Since m is the slope of the tangent then

m =1

y

a2or y1 =

m

a2

Since (x1, y1) lies on y2 = 4ax therefore

y12 = 4ax1 or 2

2

m

a4= 4ax1 x1 = 2m

a.

Substituting the values of x1 and y1 in (1), we get

y = mx +m

a........(2)

Thus, y = mx +m

ais a tangent to the parabola y2 = 4ax for all values of m , where m is the slope of the

tangent and the co-ordinates of the point of contact is

m

a2,

m

a2 .

CONIC SECTION


Two tangents can be drawn from a point P (, ) to a parabola if P lies outside theparabola :

Let the parabola be y2 = 4ax .......(1)Let P (,) be the given point

The equation of a tangent to parabola (1) is y = mx +m

a.......(2)

P( , )

If line (2) passes through P (, ), then = m+m

aor m2 – m + a = 0 .......(3)

There will be two tangents to parabola (1) from P (, ) if roots of equation (3) are real and distincti.e., D > 0 i.e. if 2 – 4a > 0 P(, ) lies outside parabola (1).We can also find the angle between two tangents from point P(,) using the formula

21

21

mm1

mmtan

3. Parametric Form :

We have to find the equation of tangent to the parabola y2 = 4ax at the point (at2, 2at) or 't'

Since the equation of tangent of the parabola y2 = 4ax at (x1, y1) isyy1 = 2a(x + x1) ........(1)

replacing x1 by at2 and y1 by 2at, then (1) becomesy(2at) = 2a(x + at2)ty = x + at2

Point of intersection of tangents at any two points on the parabola :

Let the given parabola be y2 = 4ax andtwo points on the parabola are

P = (at12, 2at1) and Q = (at2

2, 2 at2)Equation of tangents at

P (at12, 2at1) and Q(at2

2, 2at2)are t1y = x + at1

2 .........(1)

Y

X

RO

P(at , 2at )1 1

2

Q(at , 2at )2 2

2{at t , a(t + t )}1 2 1 2and t2y = x + at2

2 .........(2)Solving these equations we get

x = at1t2, y = a(t1 + t2)Thus, the co-ordinates of the point of intersection of tangents at

P(at12, 2at1) and Q(at2

2, 2at2) are R(at1t2, a(t1 + t2)).Note :

(i) The Arithmetic mean of the y-co-ordinates of P and Q

21

21 tta2

at2at.,e.i is the

y - co-ordinate of the point of intersection of tangents at P and Q on the parabola.

(ii) The Geometric mean of the x-co-ordinates of P and Q

21

22

21 tatatat.,e.i is the x co-ordinate

of the point of intersection of tangents at P and Q on the parabola.

CONIC SECTION


EQUATION OF NORMALS :

1. Point form :

Since the equation of the tangent to the parabola y2 = 4ax at (x1, y1) isyy1 = 2a(x + x1) ..........(1)

The slope of the tangent at (x1, y1) = 2a/y1.

P(x , y )1 1

T

Tangent

Normal Slope of the normal at (x1, y1) = – y1/2a

Hence the equation of normal at (x1, y1) is

y – y1 = –a2

y1 (x – x1)

2. Slope form :

The equation of normal to the parabola y2 = 4ax at (x1, y1) is

y – y1 = –a2

y1 (x – x1) .......(1)

Since m is the slope of the normal

then m = –a2

y1 or y1 = – 2am

Since (x1, y1) lies on y2 = 4ax thereforey1

2 = 4ax1 or 4a2m2 = 4ax1 x1 = am2

Substituting the values of x1 and y1 in (1) we gety + 2am = m(x – am2) ........(2)

Thus, y = mx – 2am – am3 is a normal to the parabola y2 = 4ax where m is the slope of the normal. Theco-ordinates of the point of contact are (am2, – 2am).Hence y = mx + c will be normal to parabola. If and only if c = – 2am – am3

3. Parametric form :

Equation of normal of the parabola y2 = 4ax at (x1, y1) is

y – y1 = –a2

y1 (x – x1) .........(1)

Replacing x1 by at2 and y1 by 2at then (1) becomesy – 2at = – t(x – at)2

or y = – tx + 2at + at3

CONIC SECTION


THREE SUPPLEMENTARY RESULTS:

(a) Point of intersection of normals at any two points on the parabola :

Let the points P(at12, 2at1) and Q (at2

2, 2at2) lie on the parabola y2 = 4ax

The equations of the normals at P(at12, 2at1) and Q(at2

2, 2at2) arey = –t1x + 2at1 + at1

3 .......(1)and y = – t2x + 2at2 + at2

3 .......(2)

Hence point of intersection of above normals will be obtainedby solving (1) and (2), we get

R

Q(at , 2at )2 2

AXX'

Y

Y'

P(at , 2at )1

12

x = 2a + a(t12 + t2

2 + t1t2)

y = – at1t2(t1 + t2)

If R is the point of intersection then it isR = [2a + a(t1

2 + t22 + t1t2), – at1t2 (t1 + t2)]

(b) Relation between t1 and t2 if normal at t1 meets the parabola again at t2 :

Let the parabola be y2 = 4ax, equation of normal at P(at12, 2at1) is

y = – t1x + 2at1 + at13 ........(1)

Since normal meet the parabola again at Q(at22, 2at2)

2at2 = – at1t22 + 2at1 + at2

3

2a(t2 – t1) + at1 (t22 – t1

2) = 0 a(t2 – t1) [2 + t1(t2 + t1)] = 0 a(t2 – t1) 0 ( t1 and t2 are different)

Q(at , 2at )2 2

2

AXX'

Y

Y'

P(at , 2at )1

12

2 + t1(t2 + t1) = 0

t2 = – t1 –1t

2

(c) If normal to the parabola y2 = 4ax drawn at any point (at2, 2at) meet the parabola at t3 then

t3 = –t –t

2

t2 + t t3 + 2 = 0 …(i)

It has two roots t1 & t2 . Hence there are two suchpoint P(t1) &Q(t2) on the parabola fromwhere normalsare drawn and which meet parabola at R(t3)

t1 + t2 = –t3 & t1t2 = 2

y

x

R

90°

Q

P

A

90º

Thus the line joining P(t1) & Q(t2) meet x-axis at (–2a, 0)

CONIC SECTION


CO-NORMAL POINTS :

Maximum three normals can be drawn from a point to a parabola and their feet (points where the normalmeet the parabola) are called co-normal points.Let P(h, k) be any given point and y2 = 4ax be a parabola.The equation of any normal to y2 = 4ax is

y = mx – 2am – am3

If it passes through (h, k) thenk = mh – 2am – am3

Y

Y'

AB

C

P(h, k)XX'

O

am3 + m(2a – h) + k = 0 ......(i)This is a cubic equation in m, so it has three roots, saym1, m2 and m3 . m1 + m2 + m3 = 0, …(ii)

m1m2 + m2m3 + m3m1 =a

)ha2( , …(iii)

m1m2m3 = –a

k…(iv)

Hence for any given point P(h, k), (i) has three real or imaginary roots. Corresponding to each of thesethree roots, we have each normal passing through P(h, k). Hence we have three normals PA, PB andPC drawn through P to the parabola.

PointsA, B, C in which the three normals from P(h, k) meet the parabola are called co-normal points.

Properties of co-normal points :

(1) The algebraic sum of the slopes of three concurrent normals is zero. This follows from equation (ii).

(2) The algebraic sum of ordinates of the feets of three normals drawn to a parabola from a given point iszero.Let the ordinates ofA, B, C be y1, y2, y3 respectively then

y1 = –2 am1, y2 = –2am2 and y3 = –2am3Algebraic sum of these ordinates is

y1 + y2 + y3 = –2am1 – 2am2 – 2am3= –2a(m1 + m2 + m3)= –2a × 0 {from equation (ii)}= 0

(3) If three normals drawn to any parabola y2 = 4ax from a given point (h, k) is real then h > 2a.When normals are real, then all the three roots of equation (i) are real and in that case

0mmm 23

22

21 (for any values of m1, m2, m3)

(m1 + m2 + m3)2 – 2 (m1m2 + m2m3 + m3m1) > 0

(0)2 –a

)ha2(2 > 0

h – 2a > 0or h > 2a

CONIC SECTION


(4) The centroid of the triangle formed by the feet of the three normals lies on the axis of the parabola.IfA(x1, y1), B(x2, y2) and C(x3, y3) be vertices of ABC, then its centroid is

3

yyy,

3

xxx 321321=

0,

3

xxx 321

Since y1 + y2 + y = 0 (from result-2). Hence the centroid lies on the x-axis, which is the axis of theparabola also.

Now3

xxx 321 =

3

1)amamam( 2

322

21 2

322

21 mmm

3

a

=3

a{(m1 + m2 + m3)

2 – 2(m1m2 + m2m3 + m3m1)}

=

a

ha22)0(

3

a 2=

3

a4h2

Centroid of ABC is

0,

3

a4h2

CHORD OF THE PARABOLA y2=4ax WHOSE MIDDLE POINT IS GIVEN:

Equation of the parabola is y2 = 4ax .......(1)LetAB be a chord of the parabola whose middle point is P (x1, y1).Equation of chordAB is y – y1 = m (x – x1) .......(2)where m = slope ofABLet A = (x2, y2) and B = (x3, y3).SinceAand B lie on parabola (1)

22y = 4ax2 and

23y = 4ax3

23

22 yy = 4a (x2 – x3) or

32

32

xx

yy

=

32 yy

a4

.......(3)

B(x , y )33

P(x , y )11

A(x , y )22But P(x1, y1) is the middle point ofAB y2 + y3 = 2y1

From (3),32

32

xx

yy

=

1y2

a4=

1y

a2

Slope of AB i.e., m =1y

a2.......(4)

From (2), equation of chordAB is y – y1 =1y

a2(x – x1)

or yy1 – 21y = 2ax – 2ax1 or yy1 – 2ax = 2

1y – 2ax1

or yy1 – 2a (x – x1) = 21y – 4ax1 [Subtracting 2ax1 from both sides] .......(5)

(5) is the required equation. In usual notations, equation (5) can be written as T = S1.

The same result holds true for circle, ellipse and hyperbola also.

CONIC SECTION


PAIR OF TANGENTS :

Let the parabola be y2 = 4ax .......(1)Let P(x1, y1) be a point outside the parabola.Let a chord of the parabola through the point P(x1, y1) cut the parabola at R and let Q (, ) be anarbitrary point on line PR. Let R divide PQ in the ratio : 1,

then R =

1

y,

1

x 11 .

Since R lies on parabola (1), therefore,

Q( )

RP(x , y )11

1

xaa4

1

y 12

1= 0

or ( + y1)2 – 4a( + x1) ( + 1) = 0

or (2 – 4a) 2 + 2[y1 – 2a ( + x1)] + ( 21y – 4ax1) = 0 .......(2)

Line PQ will become tangent to parabola (1) if roots of equation (2) are equal or if

4 [y1 – 2a ( + x1)]2 = 4(2 – 4a) ( 2

1y – 4ax1)

Hence, locus of Q (, ) i.e. equation of pair of tangents from P (x1, y1) is

[yy1 – 2a (x + x1)]2 = (y2 – 4ax) ( 2

1y – 4ax1)

SS1 = T2

where S, S1 and T have usual meanings.


CHORD OF CONTACT OF POINT WITH RESPECT TO A PARABOLA :

Two tangents PAand PB are drawn to parabola, then line joiningAB is called the chord of contact to theparabola with respect to point P.Let the parabola be y2 = 4ax .......(1)Let P(, ) be a point outside the parabola.Let PAand PB be the two tangents from P(, ) to parabola (1).Let A = (x1, y1) and B = (x2, y2)

Equation of the tangent PAis yy1 = 2a (x + x1) .......(2)

Equation of the tangent PB is yy2 = 2a (x + x2) .......(3)A(x , y )

11

A(x , y )22

P( )

Since lines (2) and (3) pass through P(, ), thereforey1 = 2a (a + x1) .......(4)

and y2 = 2a (a + x2) .......(5)

Now we consider the equation y = 2a (x +) ……(6)From (4) and (5), it follows that line (6) passes throughA(x1, y1) and B (x2, y2).

Hence (6) is the equation of line AB which is the chord of contact of point P(, ) with respect toparabola (1) i.e, chord of contact is y = 2a (x + )


CONIC SECTION


DIAMETER OF A PARABOLA :

Diameter of a conic is the locus of middle points of a series of its parallel chords.

Equation of diameter of a parabola :

Let the parabola be y2 = 4ax .......(1)LetABbe oneof thechords of a seriesof parallel chordshavingslope m.Let P(, ) be the middle point of chord AB, thenequation ofAB will be T = S1.or y – 2a (x + ) = 2 – 4a .......(2)

B

A

O X

Y

Slope of line (2) =

a2

but slope of line (1) i.e. lineAB is m.

a2= m or =

m

a2

Hence locus of P(,) i.e. equation of diameter(which is the locus of a series of a parallel chords having slope m) is O X

Y

m

a2y

y =m

a2.......(3)

Clearly line (3) is parallel to the axis of the parabola. Thus a diameter of a parabola is parallel to its axis.

Length of tangent, subtangent, normal and sub-normal :

Let the parabola is y2 = 4ax. Let the tangent at any pointP(x, y) meet the axis of parabola at T and G respectivelyand tangent makes an anglewith x-axis.

tan =)y,x(Pdx

dy

and PN = y N G

x

90°

P(x,y)

T

y

PT = length of tangent = PN cosec = y cosecPG = length of normal = y secTN = length of sub-tangent = PN cot = y cotNG = length of sub-normal = y tan

Properties of Parabola :

(1) Circle described on the focal length (distance) asdiameter touches the tangent at the vertex.

Equation of the circle described on SP as diameter is(x – at2) (x – a) + (y – 2at) (y – 0) = 0

x

y

P(at ,2at)2

S(a,0)OSolve it with y-axis i.e. x = 0, we gety2 – 2aty + a2t2 = 0 (y – at)2 = 0

circle touches y-axis at (0, at).

CONIC SECTION


(2) Circle described on the focal chord as diameter touches directrix

Equation of the circle described on PQ as diameter is

(x – at2)

2t

ax + (y – 2at)

t

a2y = 0

Solving it with x = – a

( – a – at2)

2t

aa + (y – 2at)

t

a2y = 0

x

y

P(at ,2at)2

S(a,0)

Qat2

2at

,–

x + a = 0 y2 – 2a

t

1t y + a2

2

t

1t

= 0

2

t

1tay

= 0

circle touches the directrix.

(3) Tangent at P is

yt = x + at2, meet x-axis at T, then T(– at2, 0)Normal at P is y + xt = 2at + at3, meet x-axis at N,then N(2a + at2, 0)

ST = SN = a + at2 = PM = PSx

P(at ,2at)2

S(a,0)

x + a = 0

q

q

NT O

y

M

PTS = TPS = TS = PS = PM TPM =

Tangent and Normal at any point P bisect the anglebetween PS and PM internally and externally. Thisproperty leads to the reflection property of parabola.

Circlecircumscribingthe triangle formedbyanytangentnormal and x-axis, has its centre at focus.If we extend MP, then from figureRPN =SPN = 90 – x

S

N

y

P RM

Thus ray parallel axis meet parabola at P and afterreflection from P it passes through the focus.

(4) The tangents at the extremities of a focal chord intersect at right angles on the directrix.

xS

y

P

90°

O90°

Q

P

Qx+a=0

CONIC SECTION


(5) The portion of tangent to the parabola intercepted between the directrix and the curve subtends a rightangle at the focus.

tangent at P(at2, 2at) is yt = x + at2 meet the directrix at x = – a

t

aat,aQ

2

and S(a, 0).

Slope at SP =aat

0at22

=

1t

t22

= m1

Slope at SQ =0a

0t

aat2

=t2

1t2

= m2.

xS

y

P

90°Q

O

m1m2 = – 1 SP SQ PSQ = 90°

(6) Tangent at P is yt = x + at2 ......(i)Line perpendicular to above line is xt + y = and passes through (a, 0) gives = at perpendicular line will be xt + y = at .....(ii)

90°

S(a, 0)x

yP(at , 2at)

2

Solve (i) and (ii), we getx = 0

i.e., these two line intersect at y-axis i.e. tangent at the vertex.The foot of the perpendicular from the focus on anytangent toa parabola lies on the tangent at vertex.

(7) Tangents and Normals at the extremities of the latus rectum of a parabola y2 = 4ax constitute a square,their points of intersection being ( a, 0) & (3a, 0).

(8) The circle circumscribing the triangle formed by any three tangents to a parabola passes through thefocus.

(9) The orthocentre of any triangle formed by three tangents to a parabola y2 = 4ax lies on the directrix &has the coordinates a , a (t1 + t2 + t3 + t1t2t3).

(10) The area of the triangle formed by three points on a parabola is twice the area of the triangle formedby the tangents at these points.

CONIC SECTION


ELLIPSE

DEFINITION :

An ellipse is the locus of the point which moves in a plane such that the ratio of its distance from a fixed

point (focus) to fixed straight line (directrix) is always constant (called eccentricity).

In thegivenfigure,S is the focusandNN'is thedirectrix.

Let P be a point on the ellipse, then

1e,ePM

PS (for ellipse)

N

M

Z S

P90°

N'Thus, we can find the equation of an ellipse when the

coordinates of its focus, equation of the directrix and

eccentricity (e) are given.

STANDARD EQUATION OF AN ELLIPSE :

Let S be the focus & ZM is the directrix of an ellipse. Draw perpendicular from S to the directrix which

meet it at Z.Amoving point is on the ellipse such that

PS = ePM

then there is point lies on the line SZ and which divide SZ

internally atAand externally atA' in the ratio of e : 1.

therefore SA = e AZ …(i)ZASCA

M

SA' = eA'Z …(ii)

Let AA' = 2a & take C as mid point of AA'

CA = CA' = a

Add (i) & (ii)

SA + SA' = e (AZ + A'Z)

AA' = e[CZ – CA + CA' + CZ]

2a = 2eCZ

CZ =e

a…(iii)

Subtract (ii) & (i), we getP(x,y)

B(0,b)

y

xA' A

(0, –b)B'

(–a,0)S' S

(ae,0)

M

ZZ'

x =ae

C(a,0)

SA' – SA = e (A'Z – AZ)

(CA' + CS) – (CA – CS)

= e [(CA' + CZ) – (CZ – CA)]

2CS = 2e CA

CS = ae …(iv)

Result (iii) & (iv) are independent of axis.

CONIC SECTION


Consider CZ line as x-axis, C as origin & perpendicular to this line & passes through C is considered as

y-axis. Let P(x, y) is a moving point, then

Bydefinitionofellipse.

PS = ePM (PS)2 = e2 (PM)2

(x – ae)2 + (y – 0)2 = e2

2

xe

a

(x – ae)2 + y2 = (a – ex)2

x2 + a2e2 – 2xae + y2 = a2 + e2x2 – 2xae x2 (1 – e2) + y2 = a2 (1 – e2)

)e1(a

y

a

x22

2

2

2

= 1 or 2

2

2

2

b

y

a

x = 1 where b2 = a2 (1 – e2)

Tracing of an ellipse :

Given ellipse is 2

2

2

2

b

y

a

x = 1 …(1)

(i) If be put y = 0 then we see that ellipse cuts x-axis at (±a, 0).

(ii) If be put x = 0 then we see that ellipse cuts y-axis at (0, ± b).

(iii) Equation of ellipse does not change when y is replaced by – y. Hence, ellipse is symmetrical about

x-axis. (Since equation contain even power of y therefore curve is symmetric about x-axis).

(iv) When x is replaced by– x, the equation of curve does not change therefore ellipse is symmetrical about

y-axis. (Since equation contain even power of x therefore curve is symmetric about y-axis).

(v) From (1), y = ± xa

a

b 2, Since y is real. a2 – x2 0 or – a x a

Also from (1), x = ±22 yb

b

a , Since x is real. b2 – y2 0 or – b y b

Hence ellipse lies entirely between the lines x = – a and x = a and the lines y = – b and y = b,

Thus an ellipse is a closed curve. Since curve is

symmetrical about both axis, therefore first of all we

draw its graph only in the first quadrant and then we

will take its image in both axis.

B (0, b)

B' (0, –b)

OA(a, 0)

B(–a, 0)

CONIC SECTION


Facts about an ellipse :

(1) By the symmetryof equation of ellipse, if we take second focus S'(–ae, 0) & second directrix x = –e

a

& perform same calculation, we get same equation of ellipse, therefore there is two focus & two directrixof an ellipse. The two foci of ellipse are (ae, 0) and (–ae, 0) and the two corresponding directrices are

lines x =e

aand x =

e

a. If focus of the ellipse is taken as (ae, 0), then corresponding directrix is x =

e

a

and if focus is (–ae, 0), then corresponding directrix is x =e

a.

(2) If equation of directrix is px + qy + r = 0 & focus is (h, k) thenits equation will be

PS2 = e2 PM2

(x – h)2 + (y – k)2 = e2 ·

2

22 qp

rqypx

P(x, y)M

S(h, k)

px + qy + r = 0

(3) Distance between foci SS' = 2ae & distance between directrix ZZ' = 2e

a.

(4) Degree of flatness of an ellipse is also called on eccentricity& written as

e =CA

CS=

vertextocentrefromDistance

focustocentrefromDistance

If e 0 b a foci becomes closer & move towards centre and ellipse becomes circle.If e 1 b 0 ellipse get thinner & thinner

(5) Two ellipse are said to be similar if theyhave same eccentricity.

(6) Distance of focus from the extremity of minor axis is equal to 'a' because a2e2 + b2 = a2

(7) Let P(x, y) be any point on the ellipse.

2

2

2

2

b

y

a

x = 1 2

2

2

2

a

x1

b

y

22

2

a

)xa)(xa(

b

y 22

2

a

N'A·AN

b

PN

y

x

P(x,y)

NC

B

A' A

B'

2

22

a

b

N'A·AN

PN

(8) Bydefinition of ellipse, the distance of anypoint P on the ellipse from focus = e (the distance of point Pfrom the corresponding directrix).

CONIC SECTION


BASIC TERMS RELATED TO AN ELLIPSE :

Let the equation of the ellipse be 2

2

2

2

b

y

a

x = 1. .......(1)

1. Centre :In the figure, C is the centre of the ellipse.All chords passing through C are called diameter and bisectedat C.

2. Foci :

S and S' are the two foci of the ellipse and theircoordinates are (ae, 0) and (–ae, 0) respectively. C

A'

L

SZ

X

x=

a/e

–

S'

Y

x=

a/e

M

M' L'

A

P

P'

Z'

N'

B'

B

N'

The line containing two foci are called the focal axisand the distance between S & S' the focal length

3. Directrices :

ZN and Z'N'are the two directrices of the ellipse and their equations are x =e

aand x =

e

arespectively..

Here Z and Z' are called foot of directrix.

4. Axes :The line segmentsA'A and B'B are called the major and minor axes respectivelyof the ellipse.The point of intersection of major and minor axis is called centre of the ellipse. Major and minor axistogether are called principal axis of ellipse.

Here Semi-major axis are CA = CA' = aand Semi-minor axis are CB = CB' = b

5. Vertex :The points where major axis meet the ellipse is called its vertices. In the given figure,A' andAare thevertices of the ellipse.

6. Ordinate and double ordinates :Let P be a point on the ellipse. From P we draw PM perpendicular to major axis of the ellipse. ProducePM to meet the ellipse at P', then PM is called an ordinate and PMP' is called the double ordinate of thepoint P.It is also defined as anychord perpendicular to major axis is called its double ordinate.

7. Latus rectum :When double ordinate passes through focus then it is called the Latus rectum.Let L'L = 2k, then LS = k so L (ae, k).Here LL' and MM' are called latus rectum.

Since L (ae, k) lies on the ellipse (1), therefore 2

2

2

22

b

k

a

ea = 1 or 2

2

b

k= 1 – e2

or k2 = b2 (1 – e2) = b2 · 2

2

a

b= 2

4

a

b[ b2 = a2 (1 – e2)]

CONIC SECTION


k =a

b2

Hence length of semi latus rectum LS =a

b2

= MS'

i.e. length of the latus rectum LL' or MM' =a

b2 2

=axismajor

)axisor(min 2

= 2a (1 – e2)

= 2e (distance from focus to the corresponding directrix).

And the end points of latus rectum are L

a

b,ae

2

, L'

a

b,ae

2

, M

a

b,ae

2

& M'

a

b,ae

2

8. Focal chord :

A chord of the ellipse passing through its focus is called a focal chord.

AUXILIARY CIRCLE AND ECCENTRIC ANGLE :A circle described on major axis as diameter is called the auxiliary circle of the given ellipse & its

equation is

x2 + y2 = a2 ... (1)

and given ellipse is 1b

y

a

x2

2

2

2

... (2)

Let Q be a point on the auxiliary circle x2 + y2 = a2 then

line through Qand perpendicular to x-axis meet the ellipse

at P then P and Q are called the CORRESPONDING POINTS

on the ellipse & the auxiliary circle respectively. Here

QOA= is called the ECCENTRIC ANGLE of the point P

on the ellipse (0 < 2 ).

Since Q lie on the circle therefore Q(a cos, a sin )

So coordinate of P(acos , y), which satisfy the equation of ellipse.

1b

y

a

cosa2

2

2

22

y = b sin

coordinate of P will be (a cos, b sin ) and this is called parametric equation of ellipse.

The equations x = a cos & y = b sin together represent the ellipse 1b

y

a

x2

2

2

2

.

Where is a eccentric angle of point P

We observe thataxismajorSemi

axisminorSemi

a

b

)QN(

)PN(

Hence “ If from each point of a circle perpendiculars are drawn upon a fixed diameter then the locus of

the points dividing these perpendiculars in a given ratio is an ellipse of which the given circle is the

auxiliarycircle”.Thisanother definitionof ellipse.

CONIC SECTION


ELLIPSE AT A GLANCE

FOCAL DISTANCE OF A POINT :

Let P(x, y) be any point on the ellipse

1b

y

a

x2

2

2

2

... (1)

Then bydefinition of ellipse, CA' A

B'

BP(x,y)

SZ' XS'

P'

Y

M'

N'

Z

M

N

T

SP = ePM = e(MT – PT) = e

x

e

a= a – ex

& S'P = ePM' = e(M'T + PT) = e

x

e

a= a + ex

Hence SP + S'P = 2a

Because of the above property, ellipse is also defined as the locus of a point which moves in a plane suchthat the sum of its distance from two fixed points (called foci) is a constant (Length of major axis).

This definition iscalled the physical definitionof the ellipse.

Hence PS + PS' = QS + QS' = TS + TS' = length of major axis

S' S

Q P

T

CONIC SECTION


TWO STANDARD FORMS OF ELLIPSE :

There are two standard forms of ellipse with centre at the origin and axes along coordinate axes. The fociof the ellipse are either on the x-axis or on the y-axis.

1. Major axis along x-axis :

The equation of this type of ellipse is of the form 1b

y

a

x2

2

2

2

, where a > b > 0 and b = a 2e1 .

For this ellipse :(i) Major axis is 2a(ii) Minor axis is 2b.(iii) Centre is (0, 0)

CA'

(-a, 0)A

(a, 0)

B'(0, –b)

B(0, b)N' N

SZ

X(0, 0)

(–ae, 0)

x=

a/e

–

(ae, 0)S'

Y

x=

a/e

Z'(iv) Vertices are (±a, 0)(v) Foci are (±ae, 0)

(vi) Equation of directrices are x = ±e

a

(vii) Equation of major axis is y = 0(viii) Equation of minor axis is x = 0

(ix) Length of latus rectum =a

b2 2

(x) Extremityof latus rectum is

a

b,ae

2

2. Major axis along y-axis :

The equation of this type of ellipse is of the form 1b

y

a

x2

2

2

2

. where 0 < a < b and a = b 2e1 .

For this ellipse :(i) Major axis is 2b(ii) Minor axis is 2a.(iii) Centre is (0, 0)(iv) Vertices are (0, ±b)(v) Foci are (0, ±be)

(vi) Equation of directrices are y = ±e

bC

A' (0, –b)

B'(–a,0)

B(a, 0)

L' M'

(0, 0)

(0, be)

S'

y=b/e

y

A (0, b)

S

x

L M

y= –b/e

(0, be)–(vii) Equation of major axis is x = 0(viii) Equation of minor axis is y= 0

(ix) Length of latus rectum =b

a2 2

(x) Extremityof latus rectum is

be,

b

a2

CONIC SECTION


Comparison chart between standard ellipse :

Basic Elements 2

2

2

2

b

y

a

x = 1

a > b b > a

Centre (0, 0) (0, 0)

Vectrex (± a, 0) (0, ± b)

Length of major axis 2a 2b

Length of minor axis 2b 2a

Foci (±ae, 0) (0, ±be)

Equation of directrixx = ±

e

ay = ±

e

b

Relation among a, b, & e b2 = a2 (1 – e2) a2 = b2 (1 – e2)

Equation of major axis y = 0 x = 0

Equation of minor axis x = 0 y = 0

Length of latus rectum

a

b2 2

b

a2 2

End of latus rectum

a

b,ae

2

be

b

a2

Focal distances of P(x1, y1) a ± ex1 b ± ey1

SP + SP’ 2a 2b

Distance between foci 2ae 2be

Distance between directrix

e

a2

e

b2

Parametric equation (a cos , b sin ) (0 < 2) (a cos , b sin )

If the equation of the ellipse is given as 1b

y

a

x2

2

2

2

& nothing is mentioned, then the rule is to assume

that a > b.

CONIC SECTION


EQUATION OF CHORD OF AN ELLIPSE :

Equation of a chord of an ellipse joining two points P() and Q() on it is equal to

2cos

2sin

b

y

2cos

a

x

(use formula of line joining points P(a cos, b sin) and Q(a cos, b sin))

If this particular chord passes through (d, 0) then we have

2cos

2cos

a

d; d

a

2cos

2cos

Using componendo and dividendo rule

da

da

2cos

2cos

2cos

2cos

or – 2cos2cos2

2sin2sin2

=

da

da

i.e.

ad

ad

2tan

2tan

if d = ae i.e. PQ is a focal chord then1e

1e

2tan

2tan

POSITION OF A POINT w.r.t. AN ELLIPSE :

Let S(x, y) 1b

y

a

x2

2

2

2

be the given ellipse

and P(x1, y1) is the given point.

(i) If S(x1, y1) > 0 then P(x1, y1) lie outside the ellipse. x

y

In

On

Out

(ii) If S(x1, y1) < 0 then P(x1, y1) lie inside the ellipse.

(iii) If S(x1, y1) = 0 then P(x1, y1) lie on the ellipse.

This result holds true for circle and parabola also.

CONIC SECTION


INTERSECTION OF A LINE AND AN ELLIPSE :

Let the equations of the line is y = mx + c ......(1)

and equation of ellipse is 2

2

2

2

b

y

a

x = 1 ......(2)

The points of intersection of the line and the ellipse canbeobtainedbysolvingthetwoequationssimultaneously.Hence by eliminating y from (1) & (2), we get

D<0D=0

D>0

2

2

2

2

b

)cmx(

a

x = 1.

i.e. (b2 + a2m2) x2 + 2a2 cmx + a2 (c2 – b2) = 0 ......(3)

Let x1, x2 be the roots of the quadratic equation (3). The line meets the ellipse in real and distinct pointsif the roots x1 and x2 are real and different. The line is a tangent to the ellipse if x1 = x2 and the line doesnot meet the ellipse if the roots x1 and x2 are imaginary.All these will be decided by the discriminant ofquadratic equation (3).

TANGENTS :

(i) Point form :

1b

yy

a

xx21

21 is tangent to the ellipse at (x1, y1).

P(x ,y )1 1tangentat P

Normalat P

Since point (x1, y1) lie on the curve therefore we can use standardsubstitution to obtain the equation of tangent.

(ii) Slope form :

Let the given line is y = mx + c and given ellipse is 2

2

2

2

b

y

a

x = 1.

If line touch is the ellipse then by solving the two equations simultaneously (by eliminating y from

(1) & (2)), we get 2

2

2

2

b

)cmx(

a

x = 1.

i.e. (b2 + a2m2) x2 + 2a2 cmx + a2 (c2 – b2) = 0 ......(3)

Since line is tangent to the ellipse therefore its D = 04a4c2m2 – 4 (b2 + a2m2) · a2(c2 – b2) = 0

or 4a2 [a2c2m2 – b2c2 – a2c2m2 + b4 + a2b2m2] = 0or b2 (–c2 + b2 + a2m2) = 0

or c2 = b2 + a2m2 or c = ± 222 bma

P( )

Q( + )

y

x

y = mx + a m + b2 2 2

y = mx – a m + b2 2 2which is the required condition of tangency.

Substituting this value of c in y = mx + c, we have

y = mx + 222 bma or y = mx – 222 bma , which are tangents to the ellipse for all values of m.

CONIC SECTION


Here ± sign represents two tangents to the ellipse having the same m, i.e. there are two tangents parallelto anygiven direction.

The equation of any tangent to the ellipse 2

2

2

2

b

)ky(

a

)hx(

= 1.

(y – k) = m (x – h) ± 222 bma

(iii) Parametric form :

1b

siny

a

cosx

is tangent to the ellipse at the point (a cos , b sin ).

NOTE:

(i) Point of intersection of the tangents at the point & is

2

2

2

2

cos

sinb,

cos

cosa can be deduced by

comparing chord joining P() and Q() with C.O.C. of the pair of tangents from R(x1, y1) on the ellipse,where

x1 =2

2

cos

cosa

; y1 =2

2

cos

sinb

P(a cos , b sin )

Q(a cos , b sin )

R

(ii) The eccentric angles of point of contact of two parallel tangents differ by. Conversely if the differencebetween the eccentric angles of two points is then the tangents at these points are parallel.

NORMALS :

(i) Point form :

Equation of the tangent to the ellipse at (x1, y1) is 21

21

b

yy

a

xx = 1.

The slope of the tangent at (x1, y1) =1

2

21

y

b

a

x

Tangent at P

Normal at P

P(x , y )1 1

Slope of the normal at (x1, y1) = 21

1 b

y

x

a

=1

21

2

xb

ya

Hence the equation of the normal at (x1, y1) is y – y1 =1

21

2

xb

ya(x – x1)

or

21

1

a

xxx

=

21

1

b

yyy

CONIC SECTION


(ii) Parametric form :

In above equation if we put x = a cos and y = b sin then we will get normal equation in parametricform. ax sec – by cosec = a2 – b2 = a²e²This is equation of normal in parametric form.

(iii) Equation of a normal in terms of its slope 'm' is y= mx 222

22

mba

m)ba(

.

DIRECTOR CIRCLE :

Locus of the point of intersection of the tangents which meet at right angles is called the Director Circle.

Let equation of any tangent is y = mx + 222 bma

If it passes through (h, k) then

k = mh ± 222 bma

(k – mh)2 = a2m2 + b2

(h2 – a2)m2 – 2khm + k2 – b2 = 0 ....(3)Equation (3) has two roots m1 & m2

m1 + m2 = 22 ah

hk2

... (4)

m1m2 = 22

22

ah

bk

... (5)

Hence passing through a given point there can be a maximum of two tangents.

If PA PB then m1m2 = –1

i.e. m1m2 = 22

22

ah

bk

= – 1

y

x

P(h,k)

A

B

90º

i.e. k2 – b2 = a2 – h2 ; i.e. x2 + y2 = a2 + b2

which is the director circle of the ellipse. Hence director circle of an ellipse is a circle whose centre is thecentre of ellipse and whose radius is the length of the line joining the ends of the major and minor axis.

Equation (3) can be used to determine the locus of the point of intersection oftwo tangents enclosing.If from anypoint P (h, k) pair of tangents are drawn to the ellipse which include an angle, then

tan =21

212

21

21

21

mm1

mm4)mm(

mm1

mm

Byputting value of m1 + m2 and m1m2 in above equation we will get the angle between pair of tangents.

CONIC SECTION


Note :If a right triangle, right angled atA circumscribes anellipse then locus of the pointAis the director circle ofthe ellipse.

CHORD OF CONTACT :

Pair of tangents drawn from outside point P (x1, y1) to the ellipse which meet it atAand B. Now linejoiningAand B is called the chord of contact of point P (x1, y1) w.r.t. the ellipse.The equation of chord of contact is

21

21

b

yy

a

xx = 1

P(x ,y )1 1 A

B

PAIR OF TANGENTS :

Pair of tangents PAand PB are drawn from outside point P (x1, y1), which is shown below. Hence jointequation of line PAand PB is given by SS1 = T2.

Here S 1b

y

a

x2

2

2

2

S1 1b

y

a

x2

21

2

21

P(x , y )1 1A

B

T 1b

yy

a

xx21

21

CHORD WITH A GIVEN MIDDLE POINT :

Here chordAB is shown in the figure whose mid point is P(x1, y1).Then equation of this chordAB is T = S1

Here S1 1b

y

a

x2

21

2

21

B

A

P(x ,y )1 1

T 1b

yy

a

xx21

21

CONIC SECTION


DIAMETER :

The locus of the middle points of a system of parallel chords

with slope 'm' of an ellipse is a straight line passing through the

centre of the ellipse, called its diameter and has the equation

y =ma

b2

2

x.

Chord of contact, pair of tangents, chord with a given middle point, pole & polar are to be interpreted as

same as they are in parabola.

Properties of the ellipse :

(1) Let the equation of an ellipse is

2

2

2

2

b

y

a

x =1 …(i)

& Normal at P(x1, y1) is

Tangent at P

y

x

p(x ,y )1 1

G SS' C

Normal at P2

1

12

1

1

b/y

yy

a/x

xx

…(ii)

The normal meet x-axis at G Put y = 0 in equation (ii) we get

CG = x =

2

22

a

bax1 = e2x1

SG = CS – CG = ae – e2x1 = e(a – ex1) = eSP

S' G S

P

Similarly S'G = eS'P

P'S

SP

P'eS

eSP

G'S

SG

PG is bisector of angle P inS'PS.

Tangent & normal at any point P bisect the external & internal angles between the focal distances

of SP & S'P.

This lead to reflection propertyof ellipse.

S' SIf incoming right ray passes through focus S'(or S),

strike the concave side of ellipse the after reflection it

will pass through other focus.

CONIC SECTION


(2) Let ellipse is 2

2

2

2

b

y

a

x = 1 …(i)

its tangent is y – mx = 222 bma …(ii)PN

N

x

y

S' C SEquation of perpendicular to above the passes

through focus (ae, 0) is

my + x = ae …(iii)

Eliminate m from (ii) &(iii)we will get focus of intersection point.

For that square & add (ii) & (iii) we will get an answer

y2 + m2x2 – 2mxy + x2 + my2 + 2mxy = a2m2 + b2 + a2e2

x2 (1 + m2) + y2 (1 + m2) = a2m2 + a2

x2 + y2 = a2 which is the auxiliarycircle

The locus of the point of intersection of feet of perpendicular from foci on any tangent to an

ellipse is the auxiliary circle.

(3) From previous equation of any tangent is mx – y + 0bma 222

SN =2

222

m1

amebma

& S'N' =

2

222

m1

amebma

SN · S'N' =)m1(

ema)bma(2

222222

= 2

222222

m1

m)ba(bma

= b2 a2e2 = a2 – b2

The product of perpendicular distance from the foci to any tangent of an ellipse is equal to square

of the semi minor axis.

(4) Equation of tangent at L

a

b,ae

2

is

1b

a

by

a

)ae(x2

2

2

xe + y = a …(i)

& equation of tangent at L'

a

b,ae

2

is

SC

y

x

L

L' x =ae

ex – y = a …(ii)

Solve (i) & (ii) we get x =e

a& y = 0 i.e. at the directrix of ellipse.

Tangents at the extremities of latus-rectum of an ellipse intersect on the corresponding directrix.

CONIC SECTION


(5) If P be any point on the ellipse with S & S as its foci then (SP) + (SP) = 2a.

(6) If thenormal at anypointPon the ellipsewith centre Cmeet the major &minoraxes in G&grespectively,

& if CF be perpendicular upon this normal, then

(i) PF. PG = b2 (ii) PF. Pg = a2

(7) The portion of the tangent to an ellipse between the point of contact &the directrix subtends a right angle

at the corresponding focus.

Equation of an ellipse referred to two perpendicular lines :

2

2

2

2

b

y

a

x = 1 is given ellipse

Let P(x, y) be any point on the ellipse, then PM = y & PN = x

y

x

P(x,y)

MO

N

above equation can be written as 1b

PM

a

PN2

2

2

2

From above we conclude that if perpendicular distances p1 & p2 of a moving point P(x, y) from two

mutually perpendicular straight lines L1 lx + my + n1 = 0 & L2 mx – ly + n2 = 0 respectively then

equation of ellipse in the plane of line will be

1b

p

a

p2

21

2

22

1b

m

nmyx

a

m

nymx

2

2

22

1

2

2

22

2

l

l

l

l b

p2

p1

p(x,y)

a

L x + my + n = 01 1l

L mx – y + n = 02 2 l

CONIC SECTION


HYPERBOLA

DEFINITION OF HYPERBOLA :

A hyperbola is the locus of a point which moves in a plane such that the ratio of its distance from a fixed

point and a given straight line is always constant.

The fixed point is called the focus, the fixed line is called the directrix and the constant ratio is called the

eccentricity of the hyperbola and denoted by e.

In the given figure, S is the focus and N'N the directrix.

Let P be any point on the hyperbola, thenPM

PS= e, e > 1.

S

PM

N

Z

90°

N

Equation of a hyperbola can be obtained if the coordinates of its focus, equation of its directrix and

eccentricityare given.

STANDARD EQUATION OF A HYPERBOLA :

Let S be the focus & ZN is the directrix of an hyperbola. Draw perpendicular from S to the directrix

which meet it at Z.Amoving point is on the hyperbola such that

PS = ePM

then there is point lies on the line SZ and which divide SZ

internally atAand externally atA' in the ratio of e : 1.

therefore SA = e AZ …(i)

SA' = eA'Z …(ii)

Let AA' = 2a & take C as mid point of AA'

x

yN' N

M'M

P(x,y)

S' A' C A S(ae,0)(–ae,0)

x=–a/e x=a/e

Z' Z

CA = CA' = a

Add (i) & (ii)

SA + SA' = e (AZ + A'Z)

(CS – CA) + (CA' + CS) = e [CA – CZ + CA' + CZ]

2CS = 2e. CA

CS = ae

Subtract (ii) & (i), we get

SA' – SA = e (A'Z – AZ)

(CA' + CS) – (CS – CA) = e [(CA' + CZ) – (CA – CZ)]

2CA = 2e · CZ CZ =e

a.

CONIC SECTION


Consider CZ line as x-axis, C as origin & perpendicular to this line & passes through C is considered as

y-axis. Now represent important parameters on coordinates plane. Let P(x, y) is a moving point, then

Bydefinitionofellipse.

PS = ePM (PS)2 = e2 (PM)2

(x – ae)2 + (y – 0)2 = e2

2

e

ax

(x – ae)2 + y2 = (a – ex)2

x2 + a2e2 – 2xae + y2 = a2 + e2x2 – 2xae x2 (1 – e2) + y2 = a2 (1 – e2)

)e1(a

y

a

x22

2

2

2

= 1 or 1

)1e(a

y

a

x22

2

2

2

where b2 = a2 (e2 – 1)

Hence equation of hyperbola is 1b

y

a

x2

2

2

2

where b2 = a2 (e2 – 1)

TRACING OF HYPERBOLA :

Equation of given hyperbola is 1b

y

a

x2

2

2

2

........(1)

(i) If we put y = 0, then we see that hyperbola cuts x-axis at (± a, 0)

(ii) If we put x = 0, then y2 = – b2 . Hence hyperbola does not cut y-axis.

(iii) When y is replaced by – y, the equation of hyperbola does not change, hence hyperbola

is symmetric about x-axis. (Since equation contain even power of y therefore curve will be

symmetric about x-axis.

(iv) When x is replaced by – x, then equation of hyperbola does not change, hence hyperbola is

symmetric about y-axis. (Since equation contain even power of x therefore curve will be

symmetric about y-axis.

(v) From (1), y =22 ax

a

b , since y is real x2 – a2 0 or x (– , – a] [a, )

curve don't lie in (– a, a)

For each x a or x – a there are two values of y symmetrically situated on both side of x-axis.

Hence, the curve denoted by (1) consist of two symmetrical branches, each extending to infinite in two

direction.

CONIC SECTION


FACTS ABOUT THE HYPERBOLA :

1b

y

a

x2

2

2

2

(1) By symmetry of equation of hyperbola, if we take second focus (–ae, 0) and second directrix x = –e

a

and perform same calculation then we get same equation of hyperbola. This suggest that their are two

foci are (ae, 0) and (– ae, 0) and the two corresponding directrices as x =e

aand x = –

e

a. If focus is

(ae, 0), then corresponding directrix is x =e

aand if focus is (– ae, 0), then corresponding directrix is

x = –e

a.

(2) By definition of hyperbola, the distance of any point P on the hyperbola from focus= e(the distance of P from the corresponding directrix)

(3) Distance between foci SS' = 2ae & distance between directrix ZZ' = 2e

a.

(4) Two hyperbola are said to be similar if theyhave same eccentricity.

(5) The hyperbola has two braches neither of them cut the y-axis (conjugate axis).

(6) Since the fundamental equation to the hyperbola only differs from that to the ellipse is having– b2 instead of b2 . It is observed that many propositions for the hyperbola are derived from those forthe ellipse bysimplychanging the sign of b2.

(7) Eccentricity e = 2

2

a

b1 . Also, b2 = a2 (e2 – 1).

The smaller the e, the smaller will be the value of b for a given a. Therefore, as e decreases for a givena, the braches of the hyperbola would be bending towards x-axis.As e increases, the braches open up.

(8) Equation of hyperbola when its transverse and conjugate axes are x and y-axes respectively is

1)1e(a

y

a

x22

2

2

2

or x2 –2

2

2

a)1e(

y

If e kept constant and a 0, then hyperbola will tend to pair of straight lines x2 – 0)1e(

y2

2

, both

passing through the origin.

Thus in the situated limiting case of a hyperbola is a pair of straight lines.

(9) Since 1b

y

a

x2

2

2

2

1a

x

b

y2

2

2

2

= 2a

)ax)(ax( (–0,0) A'

y

A(0,0) Nx

C

P(x,y)

From figure, AN = CN – CA = x – aA'N = CN + CA = x + a and PN = y

2

2

AN·N'A

)PN(= 2

2

a

b.

CONIC SECTION


TERMS RELATED TO HYPERBOLA :

(1) Centre :

In the figure, C is the centre of the ellipse.All chords passing through C are called diameter and bisectedat C.

Cx

y

S(c, 0)(–c, 0) S' A' A

B

L

L'

B'

(0,b)

(0,–b)

TA = 2a

P

P

Q

M

M' Q

(–ae,0) (ae,0)

(2) Foci :

S(ae, 0) and S'(–ae, 0) are two foci of hyperbola. Line containing the fixed points S and S' (called

Foci) is called TransverseAxis (TA) or FocalAxis and the distance between S and S' is called Focal

Length.

(3) Axes :

The line AA' is called transverse axis and the line BB' is perpendicular to it and passes through the centre

(0, 0) of the hyperbola is called conjugate axis.

The length of transverse and conjugate axes are taken as 2a and 2b respectively.

The transverse and conjugate axes together are called principal axes of hyperbola and their intersection

point is called the centre of hyperbola.

The points of intersection of the directrix with the transverse axis are known as Foot of the directrix

(Z and Z').

(4) Vertex :

The points of intersection (A,A') of the curve with the transverseaxis are calledVerticesof the hyperbola.

(5) Double ordinate :

Any chord perpendicular to the Transverse axis is called a Double Ordinate.

CONIC SECTION


(6) Latus-rectum :

When double ordinates passes through the focus of parabola then it is called the latus rectum. In thegiven figure LL' and MM' are the latus-rectums of the hyperbola.

let LL' = 2k then LS = L'S = k

Let L(ae, k) lie on the hyperbola 1b

y

a

x2

2

2

2

1b

k

a

ea2

2

2

22

x

y

NP

S' A' Z' C Z A S

x=–a/e x=a/e

Q

R

L

L'

M

M'

or k2 = b2(e2 – 1) = b2

2

2

a

b[ b2 = a2 (e2 – 1)]

k =a

b2

( k > 0)

2k =a

b2 2

= LL'

Length of latus rectum = LL' =a

b2 2

=TA

)CA( 2

= 2a(e2 – 1) = 2e

e

aae

= (2e) (distance between the focus and the foot of the corresponding directrix)End points of latus -rectums are

L

a

b,ae

2

, L'

a

b,ae

2

; M

a

b,ae

2

; M'

a

b,ae

2

respectively..

(7) Focal chord :

A chord of hyperbola passing through its focus is called a focal chord.

ECCENTRICITY :

For the hyperbola 1b

y

a

x2

2

2

2

we have

b2 = a2 (e2 – 1)

e =

2

2

a

b1 e =

2

2

)axistransverse(

)axisconjugate(1

Eccentricitydefines the curvature of the hyperbola and is mathematicallyspelled as :

e =vertextocentrefromdistance

focustocentrefromdistance

CONIC SECTION


PARAMETRIC EQUATIONS OF THE HYPERBOLA :

Let 1b

y

a

x2

2

2

2

be the hyperbola

with centre C and transverse axisA'A.

Therefore circle drawn with centre C

and segmentA'Aas a diameter is called

auxiliarycircle of the hyperbola.

X' X(–a,0)A A(a,0)

N

P(x,y)Q

Y

Y'

90°

(0,0)C

Equation of the auxiliarycircle is

x2 + y2 = a2

Let P(x, y) be any point on the hyperbola 1b

y

a

x2

2

2

2

. Draw PN perpendicular to x-axis.

Let NQ be a tangent to the auxiliary circle x2 + y2 = a2. Join CQ and let QCN =

then P and Q are the corresponding points of the hyperbola and the auxiliary circle. Here is the

eccentric angle of P. (0 < 2).

Since Q (a cos , a sin )

Now x = CN = CQ sec = sec · a

P(x, y) (a sec , y)

P lies on 1b

y

a

x2

2

2

2

1b

y

a

seca2

2

2

22

or 222

2

tan1secb

y

y = ± b tan

y = b tan (P lies in I quadrant)

The equations of x = a sec and y = tan are known as the parametric equations of the hyperbola.

1b

y

a

x2

2

2

2

Position of points Q an auxiliary circle and corresponding point P which describes the hyperbola or

shown below in the table. Here 0 < 2.

IVIV2ro2

3

IIIII2

3to

IIIIIto2

II2

to0

)tanb,seca(Psina,cosaQfromvaries

CONIC SECTION


HYPERBOLA AT A GLANCE :

Parametric coordinates x = a sec and y = b tan

Ox

y

S(ae,0)

M1

Z

A (a,0)A'(a,0)

L

L'

D1D2

M2

S'(–ae,0)

P

ax=– –eax= –e

LatusRectum

Doubleordinate

Focal axis

TA=2a

(0,–b)

(0,b)

CA=2b

Conjugate axisConjugate hyperbola

Diameter

general chord

Transverse Axis

B

B'

bae, –a

2

Z'

FOCAL DISTANCE OF A POINT ON HYPERBOLA :

X' X

Y

Y'

M' MP(x , y )1 1

SS' A' Z' C Z A

The hyperbola is 1b

y

a

x2

2

2

2

..........(1)

The foci S and S' are (ae, 0) and (– ae, 0) & corresponded directrix aree

ax and

e

ax respectively..

Let P(x1, y1) be any point on (1).

Now SP = ePM = e aexe

ax 11

CONIC SECTION


and S'P = ePM' = aexe

axe 11

Q

P

SS'

R

C

S'P – SP = (ex1 + a) – (ex1 – a) = 2a=AA' = Transverse axis

Thus hyperbola is the locus of a point which moves in aplane such that the difference of its distances from twofixed point (foci) is constant and always equal totransverse axis.Hence, in the given figurePS' – PS = QS' – QS = RS – RS' = length of transverse axis.

CHORD OF HYPERBOLA :

Chord joining two points with eccentric anglesand is given by

2cos

a

x –

2sin

b

y =

2cos

....(1)

if (1) passes through (d, 0) then

2cos

a

d =

2cos

a

d=

2sin

2sin

2cos

2cos

2sin

2sin

2cos

2cos

ad

ad

= –

2sin

2sin

2cos

2cos

2

tan2

tan

=da

da

CONJUGATE HYPERBOLA :

Corresponding to everyhyperbola there exist a hyperbola suchthat, the conjugate axis and transverse axis of one is equal tothe transverse axis and conjugate axis of other, such hyperbolasare known as conjugate to each other.

Hence for the hyperbola, 2

2

2

2

b

y

a

x = 1 .....(1)

Z

C

Z'

X' X

Y'

Y

S'(0,be)

S'(0,–be)

B'(0,–be)

S (0,be)

y = b/e

y = – b/e

the conjugate hyperbola is, 2

2

2

2

b

y

a

x = – 1 ....(2)

CONIC SECTION


Comparison between hyperbola and its conjugate hyperbola

Basic Elements Hyperbola Conjugate Hyperbola

1b

y

a

x2

2

2

2

1b

y

a

x2

2

2

2

Centre (0, 0) (0, 0)

Length of transverse axis 2a 2b

Length of conjugate axis 2b 2a

Eccentricity b2 = a2 (e2 – 1) a2 = b2 (e2 – 1)

Foci (± ae, 0) (0, ± be)

Equation of directrix x = ±e

ay = ±

e

b

Length of latus rectuma

b2 2

b

a2 2

Parametric coordinate (a sec , b tan ) (a tan , b sec )0 < < 2 0 < 2

Focal distances ex1

± a ey1

± b

Difference of focal distances 2a 2b

Equation of transverse axis y = 0 x = 0

Equation of conjugates x = 0 y = 0

Tangent at vertices x = ± a y = ± b

CONIC SECTION


Note :

(1) The foci of a hyperbola and its conjugate areconcyclic and form the vertices of a square.

O

(0,b)

(0,–b)

(0,–be ) S'2 1

S (0,be )1 2

x

y

S(ae ,0)1S'(–ae ,0)1

ConjugateHyperbola

(2) If e1& e2 are the eccentricities of the hyperbola & itsconjugate then e1

2 + e22 = 1.

RECTANGULAR HYPERBOLA :

If the lengths of transverse and conjugate axes of any hyperbola be equal then it is called rectangular orequilateral hyperbola.Since length of transverse axis and conjugate axis are same i.e. a = b

then, 2

2

2

2

b

y

a

x = 1 becomes x2 – y2 = a2.

Eccentricity, e =

2

2

a

b1 = 2 .

All the results of 2

2

2

2

b

y

a

x = 1 are applicable to the hyperbola x2 – y2 = a2 after changing b by a.

FIND ALL THE PARAMETERS OF A HYPERBOLA 1b

)ky(

a

)hx(2

2

2

2

:

When equation of the hyperbola is 1b

)y(

a

)x(2

2

2

2

This equation is the form of 1b

Y

a

X2

2

2

2

, where X = x – h and Y = y – k

(1) Length of semi-transverse axis = a, length of semi-conjugate axis = b(2) Equation of transverse axis is Y = 0, i.e, y – k = 0

Equation of conjugate axis is X = 0, i.e, x – h = 0(3) Coordinates of centre is given by X = 0 and Y = 0, i.e., x – h = 0 and y – k = 0

Therefore, centre is (h, k)

(4) Eccentricity of the hyperbola e = 2

2

a

b1

(5) Coordinates of vertices of the hyperbola are given by X = ±a, Y = 0 i.e., x – h = ± a, y – k = 0.Hence vertices are (h ± a, k).

(6) Coordinate of foci are given by X = ±ae, Y = 0i.e., x – h = ± ae, y – k = 0. Hence foci are (h ± ae, k)

CONIC SECTION


(7) Equation of directrices of the hyperbola are X = ±e

a, i.e., x – h = ±

e

a.

Hence directrices are x = h ±e

a

(8) Length of latus rectum =a

b2 2

(9) Coordinate of ends of latus rectum are given by X = ae, Y = ±a

b2

i.e. x – h = ± ae, y – h = ±a

b2

end if LR is

a

bk,aeh

2

POSITION OF A POINT WITH RESPECT TO HYPERBOLA :

Y

X

P (x , y )1 1

Q (x , y )1 2Exteriorarea

Interior area

Exteriorarea

ACA'

Interior area

L

Let S(x, y) 1b

y

a

x2

2

2

2

be the given hyperbola and P(x1, y1) is the given point.

(i) If S(x1, y1) > 0 then P(x1, y1) lie inside the ellipse.(ii) If S(x1, y1) < 0 then P(x1, y1) lie outside the ellipse.(iii) If S(x1, y1) = 0 then P(x1, y1) lie on the ellipse.

INTERSECTION OF A LINE AND A HYPERBOLA :

Let the hyperbola be 2

2

2 b

y

a

x

= 1 .......(1)

D < 0

D = 0D > 0

X

Y

and the given line be y = mx + c .......(2)The point of intersection of line and hyperbolacan be obtained by solving (1) and (2), therefore

Eliminating y from (1) and (2), we get 2

2

2

2

b

)cmx(

a

x = 1

b2x2 – a2 (mx + c)2 = a2 b2 (a2m2 – b2) x2 + 2mca2x + a2 (b2 + c2) = 0 .......(3)Above equation is a quadratic in x and gives two values of x. It shows that everystraight line will cut thehyperbola in two points, maybe real, coincident or imaginaryaccording as discriminant of (3) > , =, < 0.i.e., 4m2c2a4 – 4 (a2m2 – b2) a2 (b2 + c2) >, =, < 0or – a2m2 + b2 + c2 >, =, < 0or c2 >, =, < a2m2 – b2 .......(4)

CONIC SECTION


EQUATION OF TANGENT :

(1) Point form :

The equation of the tangent to the hyperbola 2

2

2

2

b

y

a

x = 1 at (x1, y1) is 2

121

b

yy

a

xx = 1.

Since point (x1, y1) lie on the hyperbola therefore we can use standard substitution to obtain the equationof tangent.

(2) Parametric form :

The equation of tangent to the hyperbola 2

2

2

2

b

y

a

x = 1 at (a sec , b tan ) is tan

b

ysec

a

x= 1.

Note : Point of intersection of the tangents at 1 & 2 is x = a

2cos

2cos

21

21

, y = b

2cos

2sin

21

21

(3) Slope form :

Given hyperbola is 2

2

2 b

y

a

x

= 1 ......(1)

and the given line y = mx + c touches hyperbola then solve (1) and (2) , we get

2

2

2

2

b

)cmx(

a

x = 1 or b2x2 – a2(mx + c)2 = a2b2 y

x

or (b2 – a2m2) x2 – 2a2 mcx – a2(c2 + b2) = 0 ......(3)Since line (2) will be tangent to hyperbola (1)if roots of equation (3) are equal i.e. D = 0

4a4m2c2 + 4a2(b2 – a2m2) (c2 + b2) = 0or a2m2c2 + b2c2 – a2c2m2 + b4 – a2b2m2 = 0or b2c2 + b4 – a2b2m2 = 0or c2 + b2 – a2m2 = 0

or c2 = a2m2 – b2 or c = ± 222 bma . This is the required condition of tangency

Note :

(i) Equation of any tangent to hyperbola 2

22

b

y

a

x

= 1 may be taken as y = mx + 222 bma or y = mx

– 222 bma . The co-ordinates of the points of contact are

)bma(

b,

)bma(

ma222

2

222

2

.

(ii) The equation of any tangent to the hyperbola 1b

)ky(

a

)hx(2

2

2

2

may be taken as (y – k) = m(x – h) ± 222 bma .

CONIC SECTION


Tangents drawn from outside point :

y = mx ± 222 bma is a tangent to the standard hyperbola. ....(1)

If above tangent passes through (h, k) then(k – mh)2 = a2m2 – b2

(h2 – a2)m2 – 2kmh + k2 + b2 = 0 ....(2)Above equation is quadratic in m therefore it has two roots m1 and m2.

Hence passing through a given point (h, k) there is a maximum of two tangents can be drawn to thehyperbola

therefore m1 + m2 = 22 ah

kh2

....(3)

m1m2 = 22

22

ah

bk

....(4)

Equations (3) and (4) are used to find the locus of the point of intersection of a pair of tangents whichenclose an angle.

Now tan2 = 221

212

21

)mm1(

mm4)mm(

(substituting the values of m1 + m2 and m1m2 to get the locus)If = 90° then m1m2 = – 1, hence from (4)

k2 + b2 = a2 – h2

x2 + y2 = a2 – b2 which is the equation of director circle, of the given hyperbola.

DIRECTOR CIRCLE :

The locus of the point of intersection of the tangents to the hyperbola 2

2

2

2

b

y

a

x = 1, which are

perpendicular to each other, is called the director circle.

Any tangent to hyperbola is y = mx + 222 bma ........(1)

X X

Y

Y

A CP

A

If it passes through P(h, k), then k – mh = 222 bma

k2 + m2h2 – 2mk = a2m2 – b2

m2 (h2 – a2) – 2mhk + (k2 + b2) = 0

It is quadratic in m therefore it has two roots m1 and m2. Hence two tangents (real or imaginary) can bedrawn from P(h, k).

If pair of perpendicular tangents are drawn from P(h, k) then m1 m2 = 22

22

ah

bk

= – 1.

h2 + k2 = a2 – b2

locus of P(h, k) is x2 + y2 = a2 – b2.

CONIC SECTION


Note :(i) If b2 < a2 this circle is real

(ii) If b2 = a2 the radius of the circle is zero & it reduces to a point circle at the origin. In this case the centreis the only point from which the tangents at right angles can be drawn to the curve.

(iii) If b2 > a2, the radius of the circle is imaginary, so that there is no such circle & so no tangents at rightangle can be drawn to the curve.

EQUATION OF NORMAL :

(1) Point form :

Equation of normal to a

hyperbola 2

2

2

2

b

y

a

x = 1 at a point (x1, y1) is

22

1

2

1

2

bay

yb

x

xa

yP(x ,y )1 1

Normal at P

Tangent at P

xC

(2) Parametric form :

The equation of normal at (a sec , b tan ) to the hyperbola 1b

y

a

x2

2

2

2

is

ax cos + by cot = a2 + b2

PROPERTIES :

(1) Normal and tangent at any point P(x1, y1) meet the transverse axis at G and T respectively.

The equation of normal at P on the hyperbola 1b

y

a

x2

2

2

2

is

22

1

2

1

2

bay

yb

x

xa ........(1)

The normal (1) meets the x-axis i.e. then put y = 0 in (1) we will get co-ordinates of G

i.e. G

0,x

a

)ba(12

22

or (e2x1, 0)

CG = e2x1

Now,SG = CG – CS = e2x1 – ae = e(ex1 – a) = e · SP

CONIC SECTION


Similarly, S'G = e · S'P

P'S

SP

G'S

SG

X' X

Y

Y'

P(x ,y )1 1

SS' A' C A GMT

This relation shows that the normal PGis the external bisector of the angleSPS'. The tangent PT (perpendicularto PG) is therefore the internal bisectorof the angle SPS'.The tangent & normal at any pointof a hyperbola bisect the anglebetween the focal radii.

Note : that the ellipse 1b

y

a

x2

2

2

2

and the hyperbola 1bk

y

ka

x22

2

22

2

(a > k > b > 0) are confocal

and therefore orthogonal.

(2) The tangent & normal at any point of a hyperbolabisect the angle between the focal radii. This spellsthe reflection property of the hyperbola as "Anincoming light ray " aimed towards one focus isreflected from the outer surface of the hyperbolatowards the other focus. It follows that if an ellipseand a hyperbola have the same foci, theycut at rightangles at anyof their common point.

X X

Y

S G

Y

L

A C T A S(-ae, 0) (ae, 0)

Reflected

ray

Tang

ent

Norm

al

(3) Locus of the feet of the perpendicular drawn from focus of the hyperbola 1b

y

a

x2

2

2

2

upon any tangent

is its auxiliary circle i.e. x2 + y2 = a2 & the product of the feet of these perpendiculars is b2 ·i.e., square of semi-conjugate axis.Similar propertyexists in ellipse also.

(4) The portion of the tangent between the point of contact & the directrix subtends a right angle at thecorresponding focus.

(5) The foci of the hyperbola and the points P and Q in which any tangent meets the tangents at the verticesare concyclic with PQ as diameter of the circle.

PAIR OF TANGENTS :

The combined equation of the pair of tangents drawn from a point P(x1, y1), lying outside the hyperbola

1b

y

a

x2

2

2

2

is

2

21

21

2

21

2

21

2

2

2

2

1b

yy

a

xx1

b

y

a

x1

b

y

a

x

or SS1 = T2

where S = 1b

y

a

x2

2

2

2

; S1 = 1b

y

a

x2

21

2

21 and T = 1

b

yy

a

xx21

21 .

CONIC SECTION


CHORD OF CONTACT :If the tangents from a point P(x1, y1)

to the hyperbola 1b

y

a

x2

2

2

2

touch

the hyperbola at Q and R, then theX' X

Y'

Y

C P

Q

R

Chord of contact

equation of the chord of contact QR is

1b

yy

a

xx21

21 .

Equation of the chord bisected at a given point :

The equation of the chord of the hyperbola 1b

y

a

x2

2

2

2

,

bisected at the point P (x1, y1) is

1b

yy

a

xx21

21 = 1

b

y

a

x2

21

2

21

Q(x ,y )2 2

P(x ,y )1 1

R(x ,y )3 3

or T = S1, where T = 1b

yy

a

xx21

21 ans S1 = 1

b

y

a

x2

21

2

21 .

ASYMPTOTES OF HYPERBOLA :

If the lengthof theperpendicular let fallfrom a point on a hyperbola to astraight line tends to zero as the pointonthehyperbolamovesto infinityalongthe hyperbola, then the straight line iscalled theAsymptoteof the Hyperbola.

In short asymptote is tangent tohyperbola at infinity.

Let y = mx + c is the asymptote of the

hyperbola 1b

y

a

x2

2

2

2

.

Solving these two we get the quadratic as(b2 a2m2) x2 2a2 mcx a2 (b2 + c2) = 0 ....(1)

In order that y = mx + c be an asymptote, both roots of equation (1) must approach at infinity, theconditions for which are

coeff of x2 = 0 & coeff of x = 0.

b2 a2m2 = 0 or m =a

b &

a2 mc = 0 c = 0.

equations of asymptote are 0b

y

a

x and 0

b

y

a

x .

combined equation to the asymptotes 0b

y

a

x2

2

2

2

.

As asymptotes of anyhyperbola or a curve is a straight line which touches in it two points at infinity.

CONIC SECTION


Note :

(i) If b = a, then 2

2

2 b

y

a

x

= 1 reduces x2 – y2 = a2. The asymptotes of rectangular hyperbola

x2 – y2 = a2 are y = ± x which are at right angles.

90°

B

XX'

Y

Y'

y = xy = –x

x – y = a22 2

– x + y = a22 2

A' A

B'

C

(ii) A hyperbola and its conjugate have the same asymptote.

(iii) The asymptotes pass through the centre of the hyperbola & the bisectors of the angles between theasymptotes are the axes of the hyperbola.

(iv) Asymptotes are the tangent to the hyperbola from the centre.

(v) A simple method to find the coordinates of the centre of the hyperbola is expressed as a general equationof degree 2.

i.e. let f (x, y) = 0 represents a hyperbola.

Findx

f

&

y

f

. Then the point of intersection of

x

f

= 0 &

y

f

= 0.

gives the centre of the hyperbola.

(vi) Let H 2

2

2 b

y

a

x

– 1 = 0 ; A 2

2

2 b

y

a

x

= 1 and C = 2

2

2 b

y

a

x

+ 1 = 0

be the equation of the hyperbola, asymptotes and the conjugate hyperbola respectively, then clearlyC + H = 2A i.e., all the equation different onlybyconstant term and the constant term of H,Aand C areinA.P.

(vii) The asymptotes of a hyperbola are the diagonals of the rectangle formed by the lines drawn through theextremities of each axis parallel to the other axis. Here area of rectangle is 4ab.

CONIC SECTION


PROPERTIES OF ASYMPTOTES :

(i) Perpendicular from the foci on either asymptote meet it in the same points as the corresponding directrix& the common points of intersection lie on the auxiliarycircle.

90°

B

X'

y = –x

A' A

B'

CZ' Z

K1K2

K3 K4

F

SS'

QP

Q'

P'

N

(ii) The angle between the asymptotes of 2

2

2 b

y

a

x

= 1 is 2

a

btan 1

.

and if the angle between the asymptote of a hyperbola 1b

y

a

x2

2

2

2

is 2 then e = sec.

(iii) If from any point on the asymptote a straight line be drawn perpendicular to the transverse axis, theproduct of the segments of this line, intercepted between the point & the curve is always equal to thesquare of the semi conjugate axis.

(iv) The tangent at anypoint P on a hyperbola 1b

y

a

x2

2

2

2

with centre C, meets the asymptotes in Q and R

and cuts off a CQR of constant area equal to ab from the asymptotes & the portion of the tangentintercepted between the asymptote is bisected at the point of contact.

The rectangular hyperbola xy = c2

When the centre of any rectangular hyperbola be at the origin and its asymptotes coincide with theco-ordinate axes then equation of hyperbola is xy = c2 .Here the equation of asymptotes is xy = 0 and the equation conjugate hyperbola is xy = –c2 .

Note :(i) The equation of a rectangular hyperbola having co-ordinate axes as its asymptotes is xy = c2.

If the asymptotes of a rectangular hyperbola are x =, y = then its equation is (x – ) (y – ) = c2

or xy – y – x + = 0.

(ii). Parametric equation of xy = c2 is x = ct and y =t

c.

(x, y) =

t

c,ct (t 0) is called a 't' point on the rectangular hyperbola.

CONIC SECTION


Properties of rectangular hyperbola xy = c2.

(a) Equation of a chord joining the points P (t1) & Q(t2) is x + t1t2y = c(t1 +t2) with slope m = –21tt

1.

(b) Equation of the tangent at P (x1, y1) is 2y

y

x

x

11

& at P (t) ist

x+ ty = 2c.

(c) Point of intersection of tangents at 't1' and 't2' is

2121

21

tt

c2,

tt

tct2.

(d) Equation of normal is y –t

c= t2(x – ct) or xt3 – yt – ct4 + c = 0

(e) Equation of normal (x1, y1) is xx1 – yy1 = 21

21 yx .

(f ) Chord with a given middle point as (h, k) is kx + hy = 2hk.

CONIC SECTION


EXERCISE-1 (SPECIAL DPP)

SPECIAL DPP-1 [PARABOLA]

Q.1 Which one of the following equations represented parametrically, represents equation to a parabolicprofile?

(A) x = 3 cos t ; y = 4 sin t (B) x2 2 = 2 cos t ; y = 4 cos2t

2

(C) x = tan t ; y = sec t (D) x = 1 sin t ; y = sint

2+ cos

t

2

Q.2 The locus of the point of trisection of all the double ordinates of the parabola y2 = lx is a parabola whoselatus rectum is

(A)9

l(B)

9

2l(C)

9

4l(D)

36

l

Q.3 A variable circle is drawn to touch the line 3x – 4y= 10 and also the circle x2 + y2 = 4 externally then thelocus of its centre is(A)straight line (B) circle(C) pair of real, distinct straight lines (D) parabola

Q.4 Two unequal parabolas have the same common axis which is the x-axis and have the same vertex whichis the originwith their concavities inopposite direction. If a variable line parallel to thecommon axis meetthe parabolas in P and P' the locus of the middle point of PP' is(A) a parabola (B) a circle (C) an ellipse (D) a hyperbola

Q.5 The straight line y= m(x – a) will meet the parabola y2 = 4ax in two distinct real points if(A) m R (B) m [–1, 1](C) m (– , 1] [1, )R (D) m R – {0}

Q.6 The equation of the circle drawn with the focus of the parabola (x 1)2 8 y = 0 as its centre andtouching the parabola at its vertex is(A) x2 + y2 4 y = 0 (B) x2 + y2 4 y + 1 = 0(C) x2 + y2 2 x 4 y = 0 (D) x2 + y2 2 x 4 y + 1 = 0

Q.7 The length of the intercept on yaxis cut off by the parabola, y2 5y = 3x 6 is(A) 1 (B) 2 (C) 3 (D) 5

Q.8 If the line x 1 = 0 is the directrix of the parabola y2 k x + 8 = 0, then one of the values of ' k ' is

(A)8

1(B) 8 (C) 4 (D)

4

1

CONIC SECTION


Q.9 Angle between the parabolas y2 = 4(x – 1) and x2 + 4(y – 3) = 0at the common end of their latus rectum, is(A) tan–1(1) (B) tan–11 + cot–12 + cot–13

(C) tan–1 3 (D) tan–1(2) + tan–1(3)

Q.10 Length of the latus rectum of the parabola 25 [(x 2)2 + (y 3)2] = (3x 4y + 7)2 is(A) 4 (B) 2 (C) 1/5 (D) 2/5

Q.11 PQ is a chord of parabola x2 = 4y which subtends right angle at vertex. Then locus of centroid oftriangle PSQ, where S is the focus of given parabola, is

(A) x2 = 4(y + 3) (B) x2 =3

4(y – 3) (C) x2 =

3

4(y + 3) (D) x2 =

3

4(y + 3)

Paragraph for question nos. 12 to 14

Tangents are drawn to the parabola y2 = 4x from the point P(6, 5) to touch the parabola at Q and R.C1 is a circle which touches the parabola at Q and C2 is a circle which touches the parabola at R.Both the circles C1 and C2 pass through the focus of the parabola.

Q.12 Area of thePQR equals

(A)2

1(B) 1 (C) 2 (D)

4

1

Q.13 Radius of the circle C2 is

(A) 55 (B) 105 (C) 210 (D) 210

Q.14 The common chord of the circles C1 and C2 passes through the(A) incentre of thePQR. (B) circumcentre of thePQR.(C) centroid of thePQR. (D) othocentre of thePQR.

Q.15 The locus of the mid point of the focal radii of a variable point moving on the parabola, y2 = 8x is aparabola whose(A) Latus rectum is half the latus rectum of the original parabola(B) Vertex is (1, 0)(C) Directrix is y-axis(D) Focus has the co-ordinates (2, 0)

Q.16 A variable circle is described to pass through the point (1, 0) and tangent to the curvey = tan (tan1 x). The locus of the centre of the circle is a parabola whose

(A) length of the latus rectum is 2 2

(B) axis of symmetry has the equation x + y = 1(C) vertex has the co-ordinates (3/4, 1/4)(D) directrix is x – y = 0

CONIC SECTION


Q.17 Identify the conic whose equations are given in column-I.

Column-I Column-II

(Equation of a conic) (Nature of conic)(A) xy + a2 = a(x + y) (P) Ellipse

(B) 2x2 – 72xy + 23y2 – 4x – 28y – 48 = 0 (Q) Hyperbola

(C) 6x2 – 5xy – 6y2 + 14x + 5y + 4 = 0 (R) Parabola.

(D) 14x2 – 4xy + 11y2 – 44x – 58y + 71 = 0 (S) linepair

(E) 4x2 – 4xy + y2 – 12x + 6y + 8 = 0

Q.18 A point P on a parabola y2 = 4x, the foot of the perpendicular from it upon the directrix, and the focus arethe vertices of an equilateral triangle, find the area of the equilateral triangle.


Q.1 From an external point P, pair of tangent lines are drawn to the parabola, y2 = 4x. If 1 & 2 are the

inclinations of these tangents with the axis of x such that, 1 +2 =

4, then the locus of P is :

(A) x y + 1 = 0 (B) x + y 1 = 0 (C) x y 1 = 0 (D) x + y + 1 = 0

Q.2 Maximum number of common chords of a parabola and a circle can be equal to(A) 2 (B) 4 (C) 6 (D) 8

Q.3 PN is an ordinate of the parabola y2 = 4ax (P on y2 = 4ax and N on x-axis). A straight line is drawnparallel to the axis to bisect NP and meets the curve in Q. NQ meets the tangent at the vertex A inapoint T such thatAT = kNP, then the value of k is (whereAis the vertex)(A) 3/2 (B) 2/3 (C) 1 (D) 1/3

Q.4 Minimum distance between the curves y2 = x – 1 and x2 = y – 1 is equal to

(A)4

23(B)

4

25(C)

4

27(D)

4

2

Q.5 The length of a focal chord of the parabola y2 = 4ax at a distance b from the vertex is c, then(A) 2a2 = bc (B) a3 = b2c (C) ac = b2 (D) b2c = 4a3

Q.6 The straight line joining any point P on the parabola y2 = 4ax to the vertex and perpendicular from thefocus to the tangent at P, intersect at R, then the equation of the locus of R is(A) x2 + 2y2 – ax = 0 (B) 2x2 + y2 – 2ax = 0(C) 2x2 + 2y2 – ay = 0 (D) 2x2 + y2 – 2ay = 0

CONIC SECTION


Q.7 Locus of the feet of the perpendiculars drawn from vertex of the parabola y2 = 4ax upon all such chordsof the parabola which subtend a right angle at the vertex is(A) x2 + y2 – 4ax = 0 (B) x2 + y2 – 2ax = 0 (C) x2 + y2 + 2ax = 0 (D) x2 + y2 + 4ax = 0

Q.8 The triangle PQR of area 'A' is inscribed in the parabola y2 = 4ax such that the vertex P lies at the vertexof the parabola and the base QR is a focal chord. The modulus of the difference of the ordinates of thepoints Q and R is

(A)A

a2(B)

A

a(C)

2A

a(D)

4A

a

Paragraph for Question no. 9 to 11

From the point A, common tangents are drawn to the curve C1 : x2 + y2 = 8 and C2 : y2 = 16 x.

Q.9 The y-intercept of common tangent between C1 and C2 having negative gradient, is(A) – 1 (B) – 2 (C) – 3 (D) – 4

Q.10 The radius of circle tangent to C2 at an upper end of latus rectum and passing through its focus is

(A) 23 (B) 24 (C) 26 (D) 28

Q.11 Area of quadrilateral formed by the common tangents, the chord of contact of C1 and C2with respect toA, is(A) 20 (B) 40 (C) 60 (D) 80

Q.12 Consider a circle with its centre lying on the focus of the parabola, y2 = 2 px such that it touches thedirectrix of the parabola. Then a point of intersection of the circle & the parabola is

(A)p

p2

,

(B)

pp

2,

(C)

pp

2, (D)

pp

2,

Q.13 If from the vertex of a parabola y2 = 4x a pair of chords be drawn at right angles to one another andwith these chords as adjacent sides a rectangle be made, then the locus of the further end of the rectangleis(A) an equal parabola(B) a parabola with focus at (9, 0)(C) a parabola with directrix as x 7 = 0(D) a parabola having tangent at its vertex x = 8

Q.14 The focus of the parabola is (1, 1) and the tangent at the vertex has the equation x + y = 1. Then(A) equation of the parabola is (x y)2 = 2 (x + y 1)(B) equation of the parabola is (x y)2 = 4 (x + y 1)

(C) the co-ordinates of the vertex are1

2

1

2,

(D) length of the latus rectum is 2 2

CONIC SECTION


Q.15 The parabola x = y2 + ay + b intersect the parabola x2 = y at (1, 1) at right angle.Which of the following is/are correct?(A) a = 4, b = – 4(B) a = 2, b = – 2(C) Equation of the director circle for the parabola x = y2 + ay + b is 4x + 1 = 0.

(D) Area enclosed by the parabola x = y2 + ay + b and its latus rectum is6

1.

Q.16 The straight line y + x = 1 touches the parabola(A) x2 + 4y = 0 (B) x2 x + y = 0 (C) 4x2 3x + y = 0 (D) x2 2x + 2y = 0

Q.17 If the locus of the middle points of all such chords of the parabola y2 = 8x which has a length of 2 units,has the equation y4 + ly2(2 – x) = mx – n where l, m, nN, find the value of (l + m + n)


Q.1 y-intercept of the common tangent to the parabola y2 = 32x and x2 = 108y is(A) – 18 (B) – 12 (C) – 9 (D) – 6

Q.2 The points of contact Q and R of tangent from the point P (2, 3) on the parabola y2 = 4x are

(A) (9, 6) and (1, 2) (B) (1, 2) and (4, 4) (C) (4, 4) and (9, 6) (D) (9, 6) and (4

1, 1)

Q.3 If the lines (y – b) = m1(x + a) and (y – b) = m2(x + a) are the tangents to the parabola y2 = 4ax, then(A) m1 + m2 = 0 (B) m1m2 = 1 (C) m1m2 = – 1 (D) m1 + m2 = 1

Q.4 If the normal to a parabola y2 = 4ax at P meets the curve again in Q and if PQ and the normal at Q makesangles and respectively with the x-axis then tan(tan+ tan ) has the value equal to

(A) 0 (B) – 2 (C) –2

1(D) – 1

Q.5 Equation of the other normal to the parabola y2 = 4x which passes through the intersection of those at(4, 4) and (9, 6) is(A) 5x y + 115 = 0 (B) 5x + y 135 = 0(C) 5x y 115 = 0 (D) 5x + y + 115 = 0

Q.6 The normal chord of a parabola y2 = 4ax at the point whose ordinate is equal to the abscissa, then anglesubtended bynormal chord at the focus is :

(A)

4(B) tan 1 2 (C) tan 1 2 (D)

2

CONIC SECTION


Q.7 Normals are concurrent drawn at pointsA, B, and C on the parabola y2 = 4x at P(h, k). The locus of thepoint P if the slope of the line joining the feet of two of them is 2, is

(A) x + y = 1 (B) x – y = 3 (C) y2 = 2(x – 1) (D) y2 =

2

1x2

Q.8 Tangents are drawn from the point (1, 2) on the parabola y2 = 4 x. The length , these tangents willintercept on the line x = 2 is :

(A) 6 (B) 6 2 (C) 2 6 (D) none of these

Q.9 Which one of the following lines cannot be the normals to x2 = 4y ?(A) x – y + 3 = 0 (B) x + y – 3 = 0 (C) x – 2y + 12 = 0 (D) x + 2y + 12 = 0


The line y = x – 2 cuts the parabola y2 = 8x in the points A and B. The normals drawn to the parabolaatA and B intersect at G.Aline passing through G intersects the parabola at right angles at the point C,and the tangents atAand B intersect at the point T.

Q.10 The sum of the abscissa and ordinate of the point C is(A) 0 (B) – 1 (C) 1 (D) 2

Q.11 Minimum radius of the circle passing throughAand B is

(A) 8 (B) 54 (C) 63 (D) 10

Q.12 If the two parabolas y2 = 4x and y2 = (x – k) have a common normal other than the x-axisthen k can be equal to(A) 1 (B) 2 (C) 3 (D) 4

Q.13 PQ is a double ordinate of the parabola y2 = 4ax. If the normal at P intersect the line passing through Qand parallel to axis of x at G, then locus of G is a parabola with(A) length of latus rectum equal to 4a. (B) vertex at (4a, 0).(C) directrix as the line x – 3a = 0 (D) focus as (5a, 0)

Q.14 Acircle 'S' is described on the focal chord of the parabola y2 = 4x as diameter. If the focal chord isinclined at an angle of 45° with axis of x, then which of the following is/are true?(A) Radius of the circle is 4.(B) Centre of the circle is (3, 2).(C) The line x + 1 = 0 touches the circle.(D) The circle x2 + y2 + 2x – 6y + 3 = 0 is orthogonal to 'S'.

Q.15 Let PQ be a variable normal chord of the parabola y2 = 4x whose focus is S. If the locus of the centroidof the triangle SPQ is the curve gy2 (3x – 5) = ky4 + w where g, k, w, N, find the least value of(g + k + w).

CONIC SECTION


Integer type paragraph for question nos. 16 to 18Consider a parabola y2 = 4x with vertex at A and focus at S. PQ is a chord of the parabola which isperpendicular to the tangent at point P.

Q.16 If the abscissa and the ordinate of the point P are equal then the diameter of the circumcircle of triangle

PSQ is D . Find the value of D where D N.

Q.17 If the point P varies then the locus of the middle point of the chord PQ is y2 (y2 – 2x + 4) + = 0where N. Find the value of .

Q.18 If the tangents drawn at P and Q intersect at T and the gradient of chord PQ is unity, then find thearea of triangle PTQ.


Q.1 From a variable point P on the tangent at the vertex of a parabola y2 = 2x, a line is drawn perpendicularat chord of contact. These variable lines always passes through a fixed point, where co-ordinates are

(A)

0,

2

1(B) (1, 0) (C)

0,

2

3(D) (2, 0)

Q.2 Tangents are drawn from the points on the line x y + 3 = 0 to parabola y2 = 8x. Then the variablechords of contact pass through a fixed point whose coordinates are :(A) (3, 2) (B) (2, 4) (C) (3, 4) (D) (4, 1)

Q.3 If two normals to a parabola y2 = 4ax intersect at right angles then the chord joining their feet passesthrough a fixed point whose co-ordinates are :(A) ( 2a, 0) (B) (a, 0) (C) (2a, 0) (D) none

Q.4 Theequationofastraight linepassingthrough thepoint (3,6)andcuttingthecurvey= x orthogonallyis

(A) 4x + y – 18 =0 (B) x + y – 9 = 0 (C) 4x – y – 6 = 0 (D) none

Q.5 Acircle with radius unityhas its centre on the positive y-axis. If this circle touches the parabola y = 2x2

tangentially at the points P and Q then the sum of the ordinates of P and Q, is

(A)4

15(B)

8

15(C) 152 (D) 5

Q.6 Normal to the parabola y2 = 8x at the point P (2, 4) meets the parabola again at the point Q. If C is thecentre of the circle described on PQ as diameter then the coordinates of the image of the point C in theline y = x are(A) (– 4, 10) (B) (– 3, 8) (C) (4, – 10) (D) (– 3, 10)

Q.7 C is the centre of the circle with centre (0, 1) and radius unity. P is the parabola y= ax2. The set of valuesof 'a' for which they meet at a point other than the origin, is

(A) a > 0 (B) a

2

1,0 (C)

2

1,

4

1(D)

,

2

1

CONIC SECTION


Q.8 If the locus of the middle points of the chords of the parabola y2 = 2x which touches the circlex2 + y2 – 2x – 4 = 0 is given by (y2 + 1 – x)2 = (1 + y2), then the value of is equal to(A) 3 (B) 4 (C) 5 (D) 6

Paragraph for question nos. 9 to 10Consider the parabola y2 = 8x

Q.9 Area of the figure formed by the tangents and normals drawn at the extremities of its latus rectum is(A) 8 (B) 16 (C) 32 (D) 64

Q.10 Distance between the tangent to the parabola and a parallel normal inclined at 30° with the x-axis, is

(A)3

16(B)

9

316(C)

3

2(D)

3

16

Paragraph for question nos. 11 to 13Consider the curve C : y2 – 8x – 4y + 28 = 0. Tangents TP and TQ are drawn on C at P(5, 6) andQ(5, – 2). Also normals at P and Q meet at R.

Q.11 The coordinates of circumcentre of PQR, is(A) (5, 3) (B) (5, 2) (C) (5, 4) (D) (5, 6)

Q.12 The area of quadrilateral TPRQ, is(A) 8 (B) 16 (C) 32 (D) 64

Q.13 Angle between a pair of tangents drawn at the end points of the chord y + 5t = tx + 2

of curve RtC , is

(A)6

(B)

4

(C)

3

(D)

2

Q.14 Consider the parabola y2 = 12xColumn-I Column-II

(A) Tangent and normal at the extremities of the latus rectum intersect (P) (0, 0)the x axis at T and G respectively. The coordinates of the middlepoint of T and G are

(B) Variable chords of the parabola passing through a fixed point K on (Q) (3, 0)the axis, such that sum of the squares of the reciprocals of the twoparts of the chords through K, is a constant. The coordinate of thepoint K are (R) (6, 0)

(C) All variable chords of the parabola subtending a right angle at theorigin are concurrent at the point

(D) AB and CD are the chords of the parabola which intersect at a point (S) (12, 0)E on the axis. The radical axis of the two circles described onABand CD as diameter always passes through

CONIC SECTION


Q.15 Let L1 : x + y = 0 and L2 : x – y = 0 are tangent to a parabola whose focus is S(1, 2).

If the length of latus-rectum of the parabola can be expressed asn

m(where m and n are coprime)

then find the value of (m + n).

Q.16 The normals to the parabola y2 = 4x at the points P, Q & R are concurrent at the point (15, 12). Find(a) the equation of the circle circumscribing the triangle PQR(b) the co-ordinates of the centroid of the triangle PQR.

SPECIAL DPP-5 [ELLIPSE]

Q.1 The eccentricity of the ellipse (x – 3)2 + (y – 4)2 =9

y2

is

(A)2

3(B)

3

1(C)

23

1(D)

3

1

Q.2 F1 and F2 are the two foci of the ellipse 14

y

9

x 22

. Let P be a point on the ellipse such that

21 PF2PF , where F1 and F2 are the two foci of the ellipses. The area of PF1F2 is

(A) 3 (B) 4 (C) 5 (D)2

13

Q.3 A circle has the same centre as an ellipse & passes through the foci F1 & F2 of the ellipse, such that thetwo curves intersect in 4 points. Let 'P' be any one of their point of intersection. If the major axis of theellipse is 17 & the area of the triangle PF1F2 is 30, then the distance between the foci is :(A) 11 (B) 12 (C) 13 (D) none

Q.4 Let S(5, 12) and S'(– 12, 5) are the foci of an ellipse passing through the origin.The eccentricityof ellipse equals

(A)2

1(B)

3

1(C)

2

1(D)

3

2

Q.5 The normal at a variable point P on an ellipsex

a

y

b

2

2

2

2 = 1 of eccentricitye meets the axes of the ellipse

in Q and R then the locus of the mid-point of QR is a conic with an eccentricity e such that :(A) e is independent of e (B) e = 1

(C) e = e (D) e =e

1

CONIC SECTION


Q.6 The area of the rectangle formed by the perpendiculars from the centre of the standard ellipse to thetangent and normal at its point whose eccentric angle is/4 is :

(A) 22

22

ba

ab)ba(

(B)

ab)ba(

)ba(22

22

(C)

)ba(ab

)ba(22

22

(D)

ab)ba(

ba22

22


Consider the ellipse4

y

9

x 22

= 1 and the parabola y2 = 2x. They intersect at P and Q in the first and

fourth quadrants respectively. Tangents to the ellipse at Pand Q intersect the x-axis at R and tangents tothe parabola at P and Q intersect the x-axis at S.

Q.7 The ratio of the areas of the triangles PQS and PQR, is(A) 1 : 3 (B) 1 : 2 (C) 2 : 3 (D) 3 : 4

Q.8 The area of quadrilateral PRQS, is

(A)2

153(B)

2

315(C)

2

35(D)

2

155

Q.9 The equation of circle touching the parabola at upper end of its latus rectum and passing through itsvertex, is

(A) 2x2 + 2y2 – x – 2y = 0 (B) 2x2 + 2y2 +4x –2

9y = 0

(C) 2x2 + 2y2 + x – 3y = 0 (D) 2x2 + 2y2 – 7x + y = 0

Q.10 Consider the ellipse

2

2

2

2

sec

y

tan

x= 1 where (0, /2).

Which of the followingquantities would varyasvaries?(A) degree of flatness (B) ordinate of the vertex(C) coordinates of the foci (D) length of the latus rectum

Q.11 Extremities of the latera recta of the ellipses 1b

y

a

x2

2

2

2

(a > b) having a given major axis 2a lies on

(A) x2 = a(a – y) (B) x2= a (a + y) (C) y2 = a(a + x) (D) y2 = a (a – x)

CONIC SECTION


Q.12 Column-I Column-II

(A) The eccentricityof the ellipse which meets the straight line (P)2

1

2x – 3y = 6 on the X-axis and the straight line 4x + 5y = 20on theY-axis and whose principal axes lie along the coordinate axes, is

(B) Abar of length 20 units moves with its ends on two fixed straight lines (Q)2

1

at right angles.Apoint Pmarked on the bar at a distance of 8 units fromone end describes a conic whose eccentricity is

(C) If one extremity of the minor axis of the ellipse 1b

y

a

x2

2

2

2

and the foci (R)3

5

form an equilateral triangle, then its eccentricity, is

(D) There are exactly two points on the ellipse 1b

y

a

x2

2

2

2

(S)4

7

whose distance from the centre of the ellipse are greatest and

equal to2

b2a 22 . Eccentricityof this ellipse is equal to

Q.13 Point 'O' is the centre of the ellipse with major axisAB & minor axis CD. Point F is one focus of theellipse. If OF = 6 & the diameter of the inscribed circle of triangle OCF is 2, then the productof (AB) (CD).

Q.14 Consider two concentric circles S1 : | z | = 1 and S2 : | z | = 2 on theArgand plane.Avariable parabolais drawn through the points where 'S1' meets the real axis and having arbitrary tangent drawn to 'S2'as its directrix. If the locus of the focus of the parabola is a conic C then find the area of the quadrilateralformed by the tangents at the ends of the latus-rectum of conic C.

Q.15 Tangents are drawn to the ellipse 112

y

16

x 22

from the point R 72,8 . Find the distance which

these tangents, intercept on the ordinate through nearer focus of the ellipse.

CONIC SECTION



Q.1 The y-axis is the directrix of the ellipse with eccentricitye = 1/2 and the corresponding focus is at (3, 0),equation to its auxilarycircle is(A) x2 + y2 – 8x + 12 = 0 (B) x2 + y2 – 8x – 12 = 0(C) x2 + y2 – 8x + 9 = 0 (D) x2 + y2 = 4

Q.2 Given ellipse x2 + 4y2 = 16 and parabola y2 – 4x – 4 = 0The quadratic equation whose roots are the slopes of the common tangents to parabola and ellipse, is(A) 3x2 – 1 = 0 (B) 5x2 – 1 = 0(C) 15x2 + 2x – 1 = 0 (D) 2x2 – 1 = 0

Q.3 x 2y + 4 = 0 is a common tangent to y2 = 4x &x y

b

2 2

24 = 1. Then the value of b and the other

common tangent are given by :

(A) b = 3 ; x + 2y + 4 = 0 (B) b = 3 ; x + 2y + 4 = 0

(C) b = 3 ; x + 2y 4 = 0 (D) b = 3 ; x 2y 4 = 0

Q.4 If a focal chord of an ellipse y2 = b2 (1 – x2) cuts the ellipse at the points whose eccentric angles are

12

5and

12

23, then the value of b(b < 1) is

(A)2

3(B)

3

1(C)

2

1(D)

102

5

Q.5 Consider the particle travelling clockwise on the elliptical path25

y

100

x 22

= 1. The particle leaves the

orbit at the point (–8, 3) and travels in a straight line tangent to the ellipse.At what point will the particlecross the y-axis?

(A)

3

25,0 (B)

3

23,0 (C) (0, 9) (D)

3

26,0

Q.6 The Locus of the middle point of chords of an ellipse 125

y

16

x 22

passing through P(0, 5)

is another ellipse E. The coordinates of the foci of the ellipse E, is

(A)

5

3,0 and

5

3,0 (B) (0, – 4) and (0, 1)

(C) (0, 4) and (0, 1) (D)

2

11,0 and

2

1,0

Q.7 If a quadrilateral is formed byfour tangents to the ellipse 14

y

9

x 22

is a square,

then the area of the square is equal to(A) 26 (B) 24 (C) 22 (D) 20

CONIC SECTION


Q.8 If a tangent having slope 2 of the ellipse 1b

y

a

x2

2

2

2

(a > 0, b > 0) is normal to the circle

x2 + y2 + 4x + 1 = 0, then the maximum value of ab is equal to(A) 4 (B) 3 (C) 2 (D) 1

Q.9 Tangents are drawn from the point P 2,3 to an ellipse 4x2 + y2 = 4.

Statement-1: The tangents are mutuallyperpendicular.Statement-2: The locus of the points from which mutually perpendicular tangents can be drawn to

given ellipse is x2 + y2 = 5.(A) Statement-1 is true, statement-2 is true; statement-2 is correct explanation for statement-1.(B) Statement-1 is true, statement-2 is true; statement-2 is NOT a correct explanation for statement-1.(C) Statement-1 is true, statement-2 is false.(D) Statement-1 is false, statement-2 is true.

Q.10 Statement-1: The ellipse16

x2

+9

y2

= 1 and9

x2

+16

y2

= 1 are congruent.

Statement-2: The ellipse16

x2

+9

y2

= 1 and9

x2

+16

y2

= 1 have the same eccentricity..

(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.(B)Statement-1 is true, statement-2 is trueandstatement-2 isNOT thecorrect explanationfor statement-1.(C) Statement-1 is true, statement-2 is false.(D) Statement-1 is false, statement-2 is true.

Paragraph for question nos. 11 to 14Let the two foci of an ellipse be (– 1, 0) and (3, 4) and the foot of perpendicular from the focus(3, 4) upon a tangent to the ellipse be (4, 6).

Q.1174MB The foot of perpendicular from the focus (– 1, 0) upon the same tangent to the ellipse is

(A)

5

34,

5

12(B)

3

11,

3

7(C)

4

17,2 (D) (– 1, 2)

Q.12 The equation of auxiliarycircle of the ellipse is(A) x2 + y2 – 2x – 4y – 5 = 0 (B) x2 + y2 – 2x – 4y – 20 = 0(C) x2 + y2 + 2x + 4y – 20 = 0 (D) x2 + y2 + 2x + 4y – 5 = 0

Q.13 The length of semi-minor axis of the ellipse is

(A) 1 (B) 22 (C) 17 (D) 19

Q.14 The equations of directrices of the ellipse are

(A) x – y + 2 = 0, x – y – 5 = 0 (B) x + y –2

21= 0, x + y +

2

17= 0

(C) x – y +2

3= 0, x – y –

2

5= 0 (D) x + y –

2

31= 0, x + y +

2

19= 0

CONIC SECTION


Q.15 Consider an ellipse 19

y

25

x 22

with centre C and a point P on it with eccentric angle4

. Normal

drawn at P intersects the major and minor axes inAand B respectively. N1 and N2 are the feet of theperpendiculars fromthefociS1 andS2 respectivelyonthe tangentatPandNis the footof theperpendicularfrom the centre of the ellipse on the normal at P. Tangent at P intersects the axis of x at T.

Match the entries of Column-I with the entries of Column-II.

Column-I Column-II(A) (CA)(CT) is equal to (P) 9

(B) (PN)(PB) is equal to (Q) 16

(C) (S1N1)(S2N2) is equal to (R) 17

(D) (S1P)(S2P) is equal to (S) 25

Q.16 From a variable point P a pair of tangent are drawn to the ellipse x2 + 4y2 = 4, the points of contactbeing Q and R. Let the sum of the ordinate of the points Q and R be unity. If the locus of the

point P has the equation kyy4

x 2

, then find k.

SPECIAL DPP-7 [HYPERBOLA]

Q.1 Consider the hyperbola 9x2 – 16y2 + 72x – 32y – 16 = 0. Find the following:(a) centre (b) eccentricity(c) focii (d) equation of directrix(e) length of the latus rectum (f) equation of auxilarycircle(g) equation of director circle

Q.2 Eccentricity of the hyperbola conjugate to the hyperbola 112

y

4

x 22

is

(A)3

2(B) 2 (C) 3 (D)

3

4

Q.3 Locus of the centre of the circle which touches the two circles x2 + y2 + 8x – 9 = 0and x2 + y2 – 8x + 7 = 0 externally, is

(A) x2 + y2 = 16 (B)15

y

1

x 22

= 1 (C)12

y

1

x 22

= 1 (D)15

y

1

x 22

= 1

Q.4 If the eccentricity of the hyperbola x2 y2 sec2 = 5 is 3 times the eccentricity of the ellipse

x2 sec2 + y2 = 25, then a value of is :

(A)6

(B)

4

(C)

3

(D)

2

CONIC SECTION


Q.5 The foci of the ellipse 1b

y

16

x2

22

and the hyperbola25

1

81

y

144

x 22

coincide. Then the value of b2 is

(A) 5 (B) 7 (C) 9 (D) 4

Q.6 The focal length of the hyperbola x2 – 3y2 – 4x – 6y – 11 = 0, is(A) 4 (B) 6 (C) 8 (D) 10

Q.7 The equationp29

x2

+

p4

y2

= 1 (p 4, 29) represents

(A) an ellipse if p is any constant greater than 4.(B) a hyperbola if p is any constant between 4 and 29.(C) a rectangular hyperbola if p is any constant greater than 29.(D) no real curve if p is less than 29.


Consider the conic C :12

y

16

x 22

= 1

Q.8 Equation of circle touching C at one extremityof latus-rectum and passing through centre of C is/are(A) 8x2 + 8y2 – 19x – 22y = 0 (B) 8x2 + 8y2 – 19x + 22y = 0(C) 8x2 + 8y2 + 19x – 22y = 0 (D) 8x2 + 8y2 – 19x – 22y = 0

Q.9 The equation of parabolas with same latus-rectum as conic C, is/are(A) y2 – 6x + 3 = 0 (B) y2 + 6x – 21 = 0 (C) y2 – 6x – 21 = 0 (D) y2 + 6x + 3 = 0

Q.10 If a hyperbola passes through extremities of minor axis of conic C and its transverse and conjugate axiscoincides with the minor and major axis of conic C respectively, and the product of eccentricity ofhyperbola and conic C is 1, then

(A) equation of hyperbola is12

y

36

x 22

= –1. (B) equation of hyperbola is3

y

9

x 22

= –1.

(C) focus of hyperbola is 32,0 . (D) focus of hyperbola is 34,0 .

Q.11 Statement-1: The latus rectum is the shortest focal chord in a parabola of length 4a.

Statement-2: As the lengthof a focal chordof the parabola2

2

t

1taisax4y

, which is minimum

when t = 1.(A) Statement-1 is True, Statement-2 is True ; Statement-2 is a correct explanation for Statement-1(B) Statement-1 isTrue, Statement-2 isTrue ; Statement-2 is NOT a correct explanation for Statement-1(C) Statement -1 is True, Statement -2 is False(D) Statement -1 is False, Statement -2 is True

CONIC SECTION


Q.12 Statement-1: If P(2a, 0) be any point on the axis of parabola, then the chord QPR, satisfy

222 a4

1

)PR(

1

)PQ(

1 .

Statement-2: There exists a point P on the axis of the parabola y2 = 4ax (other than vertex), such that

22 )PR(

1

)PQ(

1 = constant for all chord QPR of the parabola.

(A) Statement-1 is True, Statement-2 is True ; Statement-2 is a correct explanation for Statement-1(B) Statement-1 isTrue, Statement-2 isTrue ; Statement-2 is NOT a correct explanation for Statement-1(C) Statement -1 is True, Statement -2 is False(D) Statement -1 is False, Statement -2 is True

Q.13 Which of the following equations in parametric form can represent a hyperbolic profile, where 't' is aparameter.

(A) x =a

2t

t

1& y =

b

2t

t

1(B)

tx

a

y

b+ t = 0 &

x

a+

ty

b 1 = 0

(C) x = et + et & y = et et (D) x2 6 = 2 cos t & y2 + 2 = 4 cos2t

2

Q.14 Let p and q be non-zero real numbers. Then the equation (px2 + qy2 + r)(4x2 + 4y2 – 8x – 4) = 0represents(A) two straight lines and a circle, when r = 0 and p, q are of the opposite sign.(B) two circles, when p = q and r is of sign opposite to that of p.(C) a hyperbola and a circle, when p and q are of opposite sign and r 0.(D) a circle and an ellipse, when p and q are unequal but of same sign and r is of sign opposite to that of

p.

Q.15 Let C1 : 9x2 – 16y2 – 18x + 32y – 23 = 0 and C2 : 25x2 + 9y2 – 50 x – 18y + 33 = 0 are two conicsthen

(A) eccentricity of C1 is4

5.

(B) eccentricity of C2 is3

5.

(C) area of the quadrilateral with vertices at the foci of the conics is9

8.

(D) latus rectum of C1 is greater than latus rectum of C2.

Q.16 Solutions of the differential equation (1 – x2)dx

dy+ xy = ax where aR, is

(A) a conic which is an ellipse or a hyperbola with principal axes parallel to coordinates axes.(B) centre of the conic is (0, a)(C) length of one of the principal axes is 1.(D) length of one of the principal axes is equal to 2.

CONIC SECTION


Q.17 If is eliminated from the equationsa sec – x tan = y and b sec + y tan = x (a and b are constant)

then the eliminant denotes the equation of

(A) the director circle of the hyperbola 1b

y

a

x2

2

2

2

(B) auxiliarycircle of the ellipse 1b

y

a

x2

2

2

2

(C) Director circle of the ellipse 1b

y

a

x2

2

2

2

(D) Director circle of the circle x2 + y2 =2

ba 22 .

Q.18 Match the properties given in column-I with the corresponding curves given in the column-II.Column-I Column-II

(A) The curve such that product of the distances of anyof its tangent (P) Circlefrom two given points is constant, can be

(B) A curve for which the length of the subnormal at anyof its point is (Q) Parabolaequal to 2 and the curve passes through (1, 2), can be

(C) A curve passes through (1, 4) and is such that the segment joining (R) Ellipseanypoint P on the curve and the point of intersection of the normalat P with the x-axis is bisected by the y-axis. The curve can be (S) Hyperbola

(D) A curve passes through (1, 2) is such that the length of the normalat any of its point is equal to 2. The curve can be


(A) For an ellipse 14

y

9

x 22

with verticesAandA', tangent drawn at the (P) 2

point P in the first quadrant meets the y-axis in Q and the chordA'P meetsthe y-axis in M. If 'O' is the origin then OQ2 – MQ2 equals to

(B) If the product of the perpendicular distances from any point on the (Q) 3

hyperbola 1b

y

a

x2

2

2

2

of eccentricity e = 3 from its asymptotes

is equal to 6, then the length of the transverse axis of the hyperbola is(C) The locus of the point of intersection of the lines (R) 4

3x y 4 3 t = 0 and 3 tx + ty 4 3 = 0(where t is a parameter) is a hyperbola whose eccentricity is

(D) If F1 & F2 are the feet of the perpendiculars from the foci S1 & S2 (S) 6

of an ellipse3

y

6

x 22

=1 on the tangent at any point P on the ellipse,

then (S1F1). (S2F2) is equal to

CONIC SECTION



(A) The number of real common tangents to the circle 5x2 + 5y2 = 16 (P) 1and the hyperbola 3x2 – y2 = 48, is

(B) The number of real common normals to parabola y2 = 4x (Q) 2and the circle (x – 1)2 + (y – 1)2 = 1, is

(C) If P is any point on ellipse 1y44

x 22

whose foci are at Aand B, (R) 3

then the maximum value of (PA)(PB) equals

(D) The length of latus-rectum of the parabola defined parametrically (S) 4by equations x = cos t – sin t and y = sin 2t equals

(E) The value of a for the ellipse 2

2

2

2

b

y

a

x = 1 (a > b), if the extremities (T) 5

of the latus-rectum of the ellipse having positive ordinate lies on theparabola x2 = – 2 (y – 2), is


Q.1 The number ofpossible tangents whichcan be drawnto the curve 4x29y2 = 36,which are perpendicularto the straight line 5x + 2y10 = 0 is(A) zero (B) 1 (C) 2 (D) 4

Q.2 Locus of the point of intersection of the tangents at the points with eccentric angles and

2 on the

hyperbolax

a

y

b

2

2

2

2 = 1 is :

(A) x = a (B) y = b (C) x = ab (D) y = ab

Q.3 If2

2

cos

x–

2

2

sin

y= 1 represents family of hyperbolas where ‘’ varies then

(A) distance between the foci is constant(B) distance between the two directrices is constant(C) distance between the vertices is constant(D) distances between focus and the corresponding directrix is constant

Q.4 Number of common tangent with finite slope to the curves xy = c2 & y2 = 4ax is(A) 0 (B) 1 (C) 2 (D) 4

CONIC SECTION


Q.5 P is a point on the hyperbolax

a

y

b

2

2

2

2 = 1, N is the foot of the perpendicular from P on the transverse

axis. The tangent to the hyperbola at P meets the transverse axis at T . If O is the centre of the hyperbola,the OT. ON is equal to(A) e2 (B) a2 (C) b2 (D) b2/a2

Q.6 If x + iy= i where i = 1 and and are non zero real parameters then = constant and

= constant, represents two systems of rectangular hyperbola which intersect at an angle of

(A)6

(B)

3

(C)

4

(D)

2

Q.7 Locus of the feet of the perpendiculars drawn from either foci on a variable tangent to the hyperbola16y2 – 9x2 = 1 is(A) x2 + y2 = 9 (B) x2 + y2 = 1/9 (C) x2 + y2 =7/144 (D) x2 + y2 = 1/16

Q.8 Let the major axis of a standard ellipse equals the transverse axis of a standard hyperbola and theirdirector circles have radius equal to 2R and R respectively. If e1 and e2 are the eccentricities of theellipse and hyperbola then the correct relation is(A) 4e1

2 – e22 = 6 (B) e1

2 – 4e22 = 2 (C) 4e2

2 – e12 = 6 (D) 2e1

2 – e22 = 4

Q.9 For each positive integer n, consider the point P with abscissa n on the curve y2 – x2 = 1. If dn represents

the shortest distance from the point P to the line y = x then )d·n(Limn

n has the value equal to

(A)22

1(B)

2

1(C)

2

1(D) 0

Q.10 A tangent to the ellipse 14

y

9

x 22

with centre C meets its director circle at P and Q. Then the product

of the slopes of CP and CQ, is

(A)4

9(B)

9

4(C)

9

2(D) –

4

1

Q.11 The foci of a hyperbola coincide with the foci of the ellipse 19

y

25

x 22

. Then the equation of the

hyperbola with eccentricity2 is

(A) 14

y

12

x 22

(B) 112

y

4

x 22

(C) 3x2 – y2 + 12 = 0 (D) 9x2 – 25y2 – 225 = 0

Q.12 The graph of the equation x + y = x3 + y3 is the union of(A) line and an ellipse. (B) line and a parabola.(C) line and hyperbola. (D) line and a point.

CONIC SECTION


Paragraph for question nos. 13 to 15The graph of the conic x2 – (y– 1)2 = 1 has one tangent line with positive slope that passes through theorigin. the point of tangency being (a, b). Then

Q.13 The value of sin–1

b

ais

(A)12

5(B)

6

(C)

3

(D)

4

Q.14 Length of the latus rectum of the conic is

(A) 1 (B) 2 (C) 2 (D) none

Q.15 Eccentricityof the conic is

(A)3

4(B) 3 (C) 2 (D) none

Q.16 If the circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four points P(x1, y1), Q(x2, y2),R(x3, y3), S(x4, y4), then(A) x1 + x2 + x3 + x4 = 0 (B) y1 + y2 + y3 + y4 = 0(C) x1 x2 x3 x4 = c4 (D) y1 y2 y3 y4 = c4

Q.17 The locus of the point of intersection of those normals to the parabola x2 = 8y which are at rightangles to each other, is a parabola. Which of the following hold(s) good in respect of the locus?

(A) Length of the latus rectum is 2. (B) Coordinates of focus are

2

11,0

(C) Equation of a director circle is 2y – 11 = 0 (D) Equation of axis of symmetry is y = 0.

Q.18 Column I Column II

(A) If the co-ordinates of a point are (4 tan , 3 sec ) where is a parameter (P) 3

then the points lies on a conic whose eccentricity is equal to

(B) Tangent at any point P on an ellipse whose foci are F1, F2 meets the (Q)5

4

auxiliary circle of the ellipse at B1, B2. If F1P + F2P = 10 and(F1B1) · (F2B2) = 16, then eccentricity of the ellipse is equal to

(C) If AB is a double ordinate of a hyperbola9

y

a

x 2

2

2

= 1 (R)3

5

such that OAB is an equilateral triangle of side 2, then eccentricityof hyperbola is equal to (where O is centre of hyperbola)

(D) If the foci of ellipse 2

2

22

2

a

y

ak

x = 1 and the hyperbola 2

2

2

2

a

y

a

x = 1 (S)

5

3

coincide then one of the value of k is equal to (T)3

13

CONIC SECTION



(A) If the chord of contact of tangents from a point P to the (P) Straight lineparabola y2 = 4ax touches the parabola x2 = 4by, the locus of P is

(B) A variable circle C has the equation (Q) Circlex2 + y2 – 2(t2 – 3t + 1)x – 2(t2 + 2t)y + t = 0, where t is a parameter.The locus of the centre of the circle is

(C) The locus of point of intersection of tangents to an ellipsex

a

y

b

2

2

2

2 = 1 (R) Parabola

at two points the sum of whose eccentric angles is constant is

(D) An ellipse slides between two perpendicular straight lines. (S) HyperbolaThen the locus of its centre is

EXERCISE-2

SECTION-A [PARABOLA]Q.1 Prove that the locus of the middle point of portion of a normal to y2 = 4ax intercepted between the curve

& the axis is another parabola. Find the vertex & the latus rectum of the second parabola.

Q.2 Find the equationsof the tangents to theparabolay2 =16x, whichareparallel&perpendicular respectivelyto the line 2x – y + 5 = 0. Find also the coordinates of their points of contact.

Q.3 Find the equations of the tangents of the parabola y2 = 12x, which passes through the point (2,5).Alsofind the point of contact.

Q.4 Three normals to y² = 4x pass through the point (15, 12). Show that if one of the normals is given byy = x 3 & find the equations of the others.

Q.5 Through the vertex O of a parabola y2 = 4x , chords OP & OQ are drawn at right angles to oneanother. Show that for all positions of P, PQ cuts the axis of the parabola at a fixed point.Also find thelocus of the middle point of PQ.

Q.6 The normal at a point P to the parabola y2 = 4ax meets its axis at G. Q is another point on the parabolasuch that QG is perpendicular to the axis of the parabola. Prove that QG2 PG2 = constant.

Q.7 Find the equation of the circle which passes through the focus of the parabola x2 = 4y& touches it at thepoint (6,9).

Q.8 PQ, a variable chord of the parabola y2 = 4x subtends a right angle at the vertex. The tangents at P andQ meet at T and the normals at those points meet at N. If the locus of the mid point of TN is a parabola,then find its latus rectum.

Q.9 A tangent is drawn to the parabola y2 = 48x which makes an angle of 30º with the axis of x.Find the distance between the tangent and a parallel normal.

CONIC SECTION


Q.10 Avariable circle passes through the point A(2, 1) and touches the x-axis. Locus of the other end of thediameter throughAis a parabola.

(a) Find the length of the latus rectum of the parabola.(b) Find the coordinates of the foot of the directrix of the parabola.(c) The two tangents and two normals at the extremities of the latus rectum of the parabola constitutes a

quadrilateral. Find area of quadrilateral.

Q.11 Show that the circle through three points the normals at which to the parabola y2 = 4ax are concurrent atthe point (h, k) is 2(x2 + y2) 2(h + 2a) x ky = 0. (Remember this result)

Q.12 Tangents are drawn to the parabola y2 = 12x at the points A, B and C such that the three tangentsform a triangle PQR. If 1, 2 and 3 be the inclinations of these tangents with the axis of x such thattheir cotangents form an arithmetical progression (in the same order) with common difference 2.Find the area of the triangle PQR.

Q.13 Two tangents to the parabola y2= 8x meet the tangent at its vertex in the points P & Q. IfPQ = 4 units, prove that the locus of the point of the intersection of the two tangents is y2 = 8 (x + 2).

Q.14 A variable chord of the parabola y2 = 8x touches the parabola y2 = 2x. The the locus of the point ofintersection of the tangent at the end of the chord is a parabola. Find its latus rectum.

Q.15 PC is the normal at P to the parabola y2 = 4ax, C being on the axis. CP is produced outwards to Q sothat PQ = CP; show that the locus of Q is a parabola.

SECTION-B [ELLIPSE]Q.1 (a) Find the equation of the ellipse with its centre (1, 2), focus at (6, 2) and passing through the

point (4, 6).(b) An ellipse passes through the points (3, 1) &(2,2) &its principal axis arealong the coordinate

axes in order. Find its equation.

Q.2 If the normal at the point P() to the ellipse 15

y

14

x 22

, intersects it again at the point Q(2),

show that cos = – (2/3).

Q.3 If s, s' are the length of the perpendicular on a tangent from the foci, a, a' are those from the vertices,

c is that from the centre and e is the eccentricityof the ellipse, 1b

y

a

x2

2

2

2

, then prove that 2

2

c'aa

c'ss

= e2

Q.4 Find the equations of the lines with equal intercepts on the axes & which touch the ellipse 19

y

16

x 22

.

Q.5 Suppose x and y are real numbers and that x2 + 9y2 – 4x + 6y + 4 = 0 then find the maximum value of(4x – 9y).

Q.6 A tangent having slope3

4 to the ellipse 1

32

y

18

x 22

, intersects the axis of x & y in pointsA & B

respectively. If O is the origin, find the area of triangle OAB.

CONIC SECTION


Q.7 Find the equation of the largest circle with centre (1, 0) that can be inscribed in the ellipsex2 + 4y2 = 16.

Q.8 Let d be the perpendicular distance from the centre of the ellipse 1b

y

a

x2

2

2

2

to the tangent drawn at a

point P on the ellipse.If F1 &F2 are the two foci of the ellipse, thenshow that (PF1PF2)2 =

2

22

d

b1a4 .

Q.9 Tangents drawn from the point P(2, 3) to the circle x2 + y2 – 8x + 6y + 1 = 0 touch the circle at the

points A and B.Thecircumcircleof the PAB cuts the directorcircleofellipse

2

22

b

3y

9

)5x(

= 1

orthogonally. Find the value of b2.

Q.10 An ellipse has foci at F1(9, 20) and F2(49, 55) in the xy-plane and is tangent to the x-axis. Find the lengthof its major axis.

Q.11 A tangent to the ellipse x2 + 4y2 = 4 meets the ellipse x2 + 2y2 = 6 at P & Q. Prove that the tangents atP & Q of the ellipse x2 + 2y2 = 6 are at right angles.

Q.12 A tangent to the ellipse 1b

y

a

x2

2

2

2

touches at the point P on it in the first quadrant & meets the

coordinate axes inA& B respectively. If PdividesABin the ratio 3 : 1 reckoning from the x-axis find theequation of the tangent.

Q.13 Let Pi and Pi' be the feet of the perpendiculars drawn from foci S, S' on a tangent Ti to an ellipse

whose length of semi-major axis is 20. If

10

1i

'ii 2560)P'S()SP( , then find the value of 100e.

(Where 'e' denotes eccentricity).

Q.14 Line L1 is parallel to the line L2. Slope of L1 is 9. Also L3 is parallel to L4. Slope of L4 is25

1.

All these lines touch the ellipse 19

y

25

x 22

. Find the area of the parallelogram formed bythese lines.

Q.15 Consider the ellipse 2

2

a

x+ 2

2

b

y= 1 with centre 'O' where a > b > 0. Tangent at any point Pon the ellipse

meets the coordinate axes at X and Y and N is the foot of the perpendicular from the origin on thetangent at P. Minimum length of XYis 36 and maximum length of PN is 4.

(a) Find the eccentricityof the ellipse.(b) Find the maximum area of an isosceles triangle inscribed in the ellipse if one of its vertex coincides with

one end of the major axis of the ellipse.(c) Find the maximum area of the triangle OPN.

CONIC SECTION


SECTION-C [HYPERBOLA]

Q.1 Find the equation to the hyperbola whose directrix is 2x + y = 1, focus (1, 1) & eccentricity 3 .Find also the length of its latus rectum.

Q.2 The hyperbola 1b

y

a

x2

2

2

2

passes through the point of intersection of the lines, 7x + 13y – 87 = 0 and

5x – 8y + 7 = 0 & the latus rectum is 32 2 /5. Find 'a' & 'b'.

Q.3 For the hyperbola 125

y

100

x 22

, prove that

(i) eccentricity= 2/5 (ii) SA. SA= 25, where S & S are the foci &Ais the vertex.

Q.4 Find the centre, the foci, the directrices, the length of the latus rectum, the length & the equations of theaxes of the hyperbola 16x2 9y2 + 32x + 36y 164 = 0.

Q.5 Find the equation of the tangent to the hyperbola x2 4y2 = 36 which is perpendicular to the linex y + 4 = 0.

Q.6 Tangents are drawn to the hyperbola 3x2 2y2 = 25 from the point (0, 5/2). Find their equations.

Q.7 If 1 & 2 are the parameters of the extremities of a chord through (ae, 0) of a hyperbola

1b

y

a

x2

2

2

2

, then show that1e

1e

2tan·

2tan 21

= 0.

Q.8 Tangents are drawn from the point (, ) to the hyperbola 3x2 2y2 = 6 and are inclined at angles and to the x axis. If tan . tan = 2, prove that 2 = 22 7.

Q.9 If two points P & Q on the hyperbola 1b

y

a

x2

2

2

2

whose centre is C be such that CP is perpendicular

to CQ & a < b, then prove that2222 b

1

a

1

QC

1

PC

1 .

Q.10 Chords of the hyperbola 1b

y

a

x2

2

2

2

are tangents to the circle drawn on the line joining the foci as

diameter. Find the locus of the point of intersection of tangents at the extremities of the chords.

Q.11 Let P (a sec , b tan ) and Q (a sec , b tan ), where + =2

, be two points on the hyperbola

1b

y

a

x2

2

2

2

. If (h, k) is the point of intersection of the normals at P & Q, then find k.

Q.12 Tangent and normal are drawn at the upper end (x1, y1) of the latus rectum P with x1 > 0 and y1 > 0,

of the hyperbola 112

y

4

x 22

, intersecting the transverse axis at T and G respectively. Find the area

of the triangle PTG.

Q.13 Find the equations of the tangents to the hyperbola x2 9y2 = 9 that are drawn from(3, 2). Find the area of the triangle that these tangents form with their chord of contact.

CONIC SECTION


EXERCISE-3

SECTION-A

(JEE-ADVANCED Previous Year's Questions)

PARABOLA

Q.1 The tangent PT and the normal PN to the parabola y2 = 4ax at a point P on it meet its axis at points T andN, respectively. The locus of the centroid of the triangle PTN is a parabola whose

(A) vertex is

0,

3

a2(B) directrix is x = 0

(C) latus rectum is3

a2(D) focus is (a, 0) [JEE 2009, 4]

Q.2 LetAand B be two distinct point on the parabola y2 = 4x. If the axis of the parabola touches a circle ofradius r havingAB as its diameter, then the slope of the line joiningAand B can be

(A)r

1(B)

r

1(C)

r

2(D)

r

2[JEE 2010, 3]

Q.3(a) Let (x, y) be any point on the parabola y2 = 4x. Let P be the point that divides the line segment from(0, 0) to (x, y) in the ratio 1 : 3. Then the locus of P is(A) x2 = y (B) y2 = 2x (C) y2 = x (D) x2 = 2y

(b) Let L be a normal to the parabola y2 = 4x. If L passes through the point (9, 6), then L is given by(A) y – x + 3 = 0 (B) y + 3x – 33 = 0 (C) y + x – 15 = 0 (D) y – 2x + 12 = 0

(c) Consider the parabola y2 = 8x. Let 1 be the area of the triangle formed by the end points of its

latus rectum and the point P

2,

2

1on the parabola, and 2 be the area of the triangle formed by

drawing tangents at P and at the end points of the latus rectum. Then2

1

is [JEE 2011, 3+4+4]

Q.4 Let S be the focus of the parabola y2 = 8x and let PQ be the common chord of the circlex2 + y2 – 2x – 4y = 0 and the given parabola. The area of the triangle PQS, is [JEE 2012, 4]

Paragraph for Questions 5 and 6

Let PQ be a focal chord of the parabola y2 = 4ax. The tangents to the parabola at P and Q meet at apoint lying on the line y = 2x + a, a > 0.

Q.5 If chord PQ subtends an angle at the vertex of y2 = 4ax, then tan =

(A)3

72(B)

3

72(C)

3

5(D)

3

5

CONIC SECTION


Q.6 Length of chord PQ is(A) 7a (B) 5a (C) 2a (D) 3a

[JEE 2013, 3+3]

Q.7 A line L : y = mx + 3 meets y-axis at E(0, 3) and the arc of the parabola y2 = 16x, 0 y 6 at the pointF(x0, y0). The tangent to the parabola at F(x0, y0) intersects the y-axis at G(0, y1). The slope m of theline Lis chosen such that the area of the triangle EFG has a local maximum.

Match List-I with List-II and select the correct answer using the code given below the lists:List I List II

P. m = 1. 1/2Q. Maximum area ofEFG is 2. 4R. y0 = 3. 2S. y1 = 4. 1

Codes:P Q R S

(A) 4 1 2 3(B) 3 4 1 2(C) 1 3 2 4(D) 1 3 4 2 [JEE 2013, 3]

Q.8 The common tangents to the circle x2 + y2 = 2 and the parabola y2 = 8x touch the circle at the points P,Q and the parabola at the points R, S. Then the area of the quadrilateral PQRS is(A) 3 (B) 6 (C) 9 (D) 15

[JEE Adv. 2014, 3]

Paragraph For Questions 9 and 10Let a, r, s, t be non zero real numbers. Let P(at2,2at), Q, R(ar2,2ar) and S(as2,2as) be distinct points onthe parabola y2 = 4ax. Suppose that PQ is the focal chord and lines QR and PK are parallel, where Kis the point (2a, 0).

Q.9 The value of r is

(A)t

1(B)

t

1t2 (C )

t

1(D)

t

1t2

Q.10 If st = 1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is

(A) 3

22

t2

)1t( (B) 3

22

t2

)1t(a (C) 3

22

t

)1t(a (D) 3

22

t

)2t(a

[JEE Adv. 2014, 3 + 3]

Q.11 If the normals of the parabola y2 = 4x drawn at the end points of its latus rectum are tangents to the circle(x – 3)2 + (y + 2)2 = r2, then the value of r2 is [JEE Adv. 2015, 4]

Q.12 Let the curve C be the mirror image of the parabola y2 = 4x with respect to the line x + y + 4 = 0. IfAand B are the points of intersection of C with the line y = – 5, then the distance betweenAand B is

[JEE Adv. 2015, 4]

CONIC SECTION


Q.13 Let P and Q be distinct points on the parabola y2 = 2x such that a circle with PQ as diameter passesthrough the vertex O of the parabola. If P lies in the first quadrant and the area of the triangleOPQ

is 23 , then which of the following is(are) the coordinates of P?

(A) 22,4 (B) 23,9 (C)

2

1,

4

1(D) 2,1

[JEE Adv. 2015, 4]

Q.14 The circle C1 : x2 + y2 = 3, with centre at O, the parabola x2 = 2y at the point P in the first quadrant. Letthe tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively. Suppose

C2 and C3 have equal radii 2 3 and centres Q2 and Q3, respectively. If Q2 and Q3 lie on the y-axis,

then(A) Q2Q3 = 12

(B) R2R3 = 4 6

(C) area of the triangle OR2R3 is 6 2

(D) area of the triangle PQ2Q3 is 4 2[JEE Adv. 2016, 4]

Q.15 Let P be the point on the parabola y2 = 4x. Which is at the shortest distance from the center S of thecircle x2 + y2 – 4x – 16y+ 64 = 0. Let Q be the point on the circle dividing the line segment SPinternally.Then

(A) SP = 2 5

(B) SQ : QP = ( 5 + 1) : 2

(C) the x-intercept of the normal to the parabola at P is 6

(D) the slope of the tangent to the circle at Q is2

1[JEE Adv. 2016, 4]

Q.16 If a chord, which is not a tangent, of the parabola y2 = 16x has the equation 2x + y = p, and midpoint(h, k), then which of the following is(are) possible value(s) of p, h and k?(A) p = 2, h = 3, k = – 4 (B) p = 5, h = 4, k = – 3(C) p = – 1, h = 1, k = – 3 (D) p = – 2, h = 2, k = – 4

[JEE (Advanced) 2017, 4]

ELLIPSE

Q.1 Let P(x1, y1) and Q(x2, y2), y1 < 0, y2 < 0, be the end points of the latus rectum of the ellipsex2 + 4y2 = 4. The equations of parabolas with latus rectum PQ are

(A) x2 + 32 y = 3 + 3 (B) x2 – 32 y = 3 + 3

(C) x2 + 32 y = 3 – 3 (D) x2 – 32 y = 3 – 3 [JEE 2008, 4]

CONIC SECTION


Q.2(i)The line passing through the extremityAof the major axis and extremityB of the minor axis of the ellipsex2 + 9y2 = 9 meets its auxiliary circle at the point M. Then the area of the triangle with vertices atA, Mand the origin O is

(A)10

31(B)

10

29(C)

10

21(D)

10

27

(ii) The normal at a point P on the ellipse x2 + 4y2 = 16 meets the x-axis at Q. If M is the mid point of the linesegment PQ, then the locus of M intersects the latus rectums of the given ellipse at the point

(A)

7

2,

2

53(B)

4

19,

2

53(C)

7

1,32 (D)

7

34,32

(iii) In a triangleABC with fixed base BC, the vertex Amoves such that cos B + cos C = 4 sin2

2

A.

If a, b and c denote the lengths of the sides of the triangle opposite to the anglesA, Band C, respectively,then(A) b + c = 4a (B) b + c = 2a(C) locus of pointAis an ellipse. (D) locus of pointAis a pair of straight lines.

[JEE 2009, 3+3+4]

Comprehension (3 question together)

Q.3 Tangents are drawn from the point P(3, 4) to the ellipse 14

y

9

x 22

touching the ellipse at pointsAA

and B.

(i) The coordinates ofAand B are

(A) (3, 0) and (0, 2) (B)

5

8,

5

9and

15

1612,

5

8

(C) )2,0(and15

1612,

5

8

(D)

5

8,

5

9and0,3

(ii) The orthocenter of the triangle PAB is

(A)

7

8,5 (B)

8

25,

5

7(C)

5

8,

5

11(D)

5

7,

25

8

(iii) The equation of the locus of the point whose distances from the point Pand the lineAB are equal, is(A) 9x2 + y2 – 6xy – 54x – 62y + 241 = 0 (B) x2 + 9y2 + 6xy – 54x + 62y – 241 = 0(C) 9x2 + 9y2 – 6xy – 54x – 62y – 241 = 0 (D) x2 + y2 – 2xy + 27x + 31y – 120 = 0

[JEE 2010, 3+3+3]

CONIC SECTION


Q.4 The ellipse E1 :4

y

9

x 22

= 1 is inscribed in a rectangle R whose sides are parallel to the

coordinate axes.Another ellipse E2 passing through the point (0, 4) circumscribes the rectangle R.The eccentricity of the ellipse E2 is

(A)2

2(B)

2

3(C)

2

1(D)

4

3[JEE 2012, 3]

Q.5 A vertical line passing through the point (h, 0) intersects the ellipse 13

y

4

x 22

at the points P and Q.

Let the tangents to the ellipse at P and Q meet at the point R. If (h) = area of the triangle PQR,

1 = )h(max1h21

and 2 = )h(min

1h21

, then

21 85

8 = [JEE 2013, 4]

Q.6 List-I List-II

P. Let y(x) = cos (3cos–1x), x [–1, 1], x ±2

3. 1. 1

Then

dx

)x(dyx

dx

)x(yd)1x(

)x(y

12

22

equals

Q. LetA1,A2, …….,An (n > 2) be the vertices of a regular polygon of 2. 2

n sides with its centre at the origin. Let ka be the position vector of

the point Ak, k = 1, 2, ……., n. If

1n

1k1kk )aa( =

1n

1k1kk )a·a( ,

then the minimumvalueof n is

R. If the normal from the point P (h, 1) on the ellipse3

y

6

x 2

= 1 3. 8

is perpendicular to the line x + y = 8, then the value of h is

S. Number of positive solutions satisfying the equation 4. 9

1x4

1tan

1x2

1tan 11 =

2

1

x

2tan is

Codes:P Q R S

(A) 4 3 2 1(B) 2 4 3 1(C) 4 3 1 2(D) 2 4 1 3 [JEE Adv. 2014, 3]

CONIC SECTION


Q.7 Suppose that the foci of the ellipse5

y

9

x 22

= 1 are (f1, 0) and (f2, 0) where f1 > 0 and f2 < 0. Let P1

and P2 be two parabolas with a common vertex at (0, 0) and with foci at (f1, 0) and (2f2, 0), respectively.Let T1 be a tangent to P1 which passes through (2f2, 0) and T2 be a tangent to P2 which passes through

(f1, 0). If m1 is the slope of T1 and m2 is the slope of T2, then the value of

2

221

mm

1is

[JEE Adv. 2015, 4]

Q.8 Let E1 and E2 be two ellipses whose centers are at the origin. The major axes of E1 and E2 lie along thex-axis and the y-axis, respectively. Let S be the circle x2 + (y – 1)2 = 2. The straight line x + y = 3

touches the curves S, E1 and E2 at P, Q and R, respectively. Suppose that PQ = PR =3

22. If e1 and

e2 are the eccentricities of E1 and E2, respectively, then the correct expression(s) is (are)

(A)40

43ee 2

221 (B)

102

7ee 21 (C)

8

5ee 2

221 (D)

4

3ee 21

[JEE Adv. 2015, 4]Paragraph for Questions 9 & 10

Let F1 (x1, 0) and F2 (x2, 0), for x1 < 0 and x2 > 0, be the foci of the ellipse8

y

9

x 22

= 1. Suppose a

parabola having vertex at the origin and focus at F2 intersect the ellipse at point M in the first quadrantand at point N in the fourth quadrant.

Q.9 The orthocentre of the triangle F1MN is

(A)

0,

10

9(B)

0,

3

2(C)

0,

10

9(D)

6,

3

2

Q.10 If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axisat Q, then the ratio of area of the triangle MQR to area of the quadrilateral MF1NF2 is(A) 3 : 4 (B) 4 : 5 (C) 5 : 8 (D) 2 : 3

[JEE Adv. 2016, 3]

Q.11 Consider two straight lines, each of which is tangent to both the circle x2 + y2 =2

1and the parabola

y2 = 4x. Let these lines intersect at the point Q. Consider the ellipse whose centre is at the origin O(0, 0)

and whose semi-major is OQ. If the length of the minor axis of this ellipse is 2 , then which of thefollowing statement(s) is(are) TRUE ?

(A) For the ellipse, the eccentricity is2

1and the length of the latus rectum is 1

(B) For the ellipse, the eccentricity is2

1and the length of the latus rectum is

2

1

(C) The area of the region bounded by the ellipse between the lines x =2

1and x = 1 is

24

1(– 2)

(D) The area of the region bounded by the ellipse between the lines x =2

1and x = 1 is

16

1(– 2)


CONIC SECTION


HYPERBOLA

Q.1(i) Let a and b be non-zero real numbers. Then, the equation (ax2 + by2 + c) (x2 – 5xy + 6y2) = 0represents(A) four straight lines, when c = 0 and a, b are of the same sign.(B) two straight lines and a circle, when a = b, and c is of sign opposite to that of a.(C) two straight lines and a hyperbola, when a and b are of the same sign and c is of sign opposite to that

of a(D) a circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a.

(ii) Consider a branch of the hyperbola, x2 – 2y2 – x22 – y24 – 6 = 0 with vertex at the pointA. Let

B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the pointA,then the area of the triangleABC is

(A)3

21 (B) 1

2

3 (C)

3

21 (D) 1

2

3

[JEE 2008, 3+3]Q.2(i) The locus of the orthocentre of the triangle formed bythe lines

(1 + p)x – py + p(1 + p) = 0, (1 + q)x – qy + q(1 + q) = 0 and y = 0, where p q, is(A) a hyperbola (B) a parabola (C) an ellipse (D) a straight line

(ii) An ellipse intersects the hyperbola 2x2 – 2y2 = 1 orthogonally.The eccentricityof the ellipse is reciprocalof that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then(A) Equation of ellipse is x2 + 2y2 = 2(B) The foci of ellipse are (±1, 0)(C) Equation of ellipse is x2 + 2y2 = 4

(D) The foci of ellipse are (± 2 , 0)

(iii) Match the conics in Column I with the statements/expressions in Column II.

Column I Column II

(A) Circle (p) The locus of the point (h, k) for which the line hx + ky = 1touches the circle x2 + y2 = 4

(B) Parabola (q) Points z in the complex plane satisfying |z + 2| – |z – 2| = ± 3

(C) Ellipse (r) Points of the conic have parametric representation

(D) Hyperbola22

2

t1

t2y,

t1

t13x

(s) The eccentricity of the conic lies in the interval 1 x <

(t) Points z in the complex plane satisfying Re(z + 1)2 = |z|2 + 1

[JEE 2009, 3+4 + (2+2+2+2)]

CONIC SECTION


Comprehension (2 question together)

Q.3(a) The circle x2 + y2 – 8x + 0 and hyperbola 14

y

9

x 22

intersect at the pointsAand B.

(i) Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

(A) 020y5x2 (B) 04y5x2

(C) 3x – 4y + 8 = 0 (D) 4x – 3y + 4 = 0

(ii) Equation of the circle withAB as its diameter is(A) x2 + y2 – 12x + 24 = 0 (B) x2 + y2 + 12x + 24 = 0(C) x2 + y2 + 24x – 12 = 0 (D) x2 + y2 – 24x – 12 = 0

(b) The line 2x + y = 1 is tangent to the hyperbola 1b

y

a

x2

2

2

2

. If this line passes through the point of

intersection of the nearest directrix and the x-axis, then the eccentricityof the hyperbola, is[JEE 2010, 3+3+3]

Q.4(a)Let P (6, 3) be a point on the hyperbola 1b

y

a

x2

2

2

2

. If the normal at the point P intersects the x-axis at

(9, 0), then the eccentricity of the hyperbola is

(A)2

5(B)

2

3(C) 2 (D) 3

(b) Let the eccentricity of the hyperbola 1b

y

a

x2

2

2

2

be reciprocal to that of the ellipse x2 + 4y2 = 4.

If the hyperbola passes through a focus of the ellipse, then

(A) the equation of the hyperbola is 12

y

3

x 22

.

(B) a focus of the hyperbola is (2, 0).

(C) the eccentricity of the hyperbola is3

5.

(D) the equation of the hyperbola is x2 – 3y2 = 3. [JEE 2011, 3+4]

Q.5 Tangents are drawn to the hyperbola 14

y

9

x 22

, parallel to the straight line 2x – y = 1. The points

of contact of the tangent on the hyperbola are

(A)

2

1,

22

9(B)

2

1,

22

9(C) 22,33 (D) 22,33

[JEE 2012, 4]

CONIC SECTION


Q.6 Consider the hyperbola H : x2 – y2 = 1 and a circle S with center N(x2, 0). Suppose that H and S toucheach other at a point P(x1, y1) with x1 > 1 and y1 > 0. The common tangent to H and S at P intersects thex-axis at point M. If (l, m) is the centroid of the trianglePMN, then the correct expression(s) is(are)

(A)211 x3

11

dx

d

lfor x1 > 1 (B) 1xfor

1x3

x

dx

dm1

21

1

1

(C) 1xforx3

11

dx

d12

11

l

(D) 0yfor3

1

dy

dm1

1

[JEE Adv. 2015, 4]

Q.7 If 2x – y + 1 = 0 is a tangent to the hyperbola16

y

a

x 2

2

2

= 1, then which of the following CANNOT be

sides of a right angled triangle?(A) 2a, 4, 1 (B) a, 4, 1 (C) a, 4, 2 (D) 2a, 8, 1


Answer Q.8, Q.9 and Q.10 byappropriatelymatching the information given in the three columns of thefollowingtable.

1ma

1,

1ma

ma)S(1mamxy)iv(ayax)IV(

1ma

1,

1ma

ma)R(1mamxy)iii(ax4y)III(

1m

a,

1m

ma)Q(1mamxy)ii(ayax)II(

m

a2,

m

a)P(axmmy)i(ayx)I(

3Column2Column1Column

2222

2222222

2222

2222

22

22222

2

2222

Column 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact,respectively.

Q.8 The tangent to a suitable conic (Column 1) at

2

1,3 is found to be 3 x + 2y = 4, then which of the

followingoptions is the onlyCORRECT combination?(A) (IV) (iii) (S) (B) (II) (iii) (R) (C) (IV) (iv) (S) (D) (II) (iv) (R)

Q.9 If a tangent to a suitable conic (Column 1) is found to be y= x + 8 and its point of contact is (8, 16), thenwhich of the following options is the onlyCORRECT combination?(A) (III) (i) (P) (B) (I) (ii) (Q) (C) (II) (iv) (R) (D) (III) (ii) (Q)

CONIC SECTION


Q.10 For a = 2 , if a tangent is drawn to a suitable conic (Column 1) at the point of contact (– 1, 1), thenwhich of the following options is the onlyCORRECT combination for obtaining its equation?(A) (II) (ii) (Q) (B) (I) (i) (P) (C) (I) (ii) (Q) (D) (III) (i) (P)

[JEE (Advanced) 2017, 3 + 3 + 3]

Q.11 Let H :2

2

2

2

b

y

a

x = 1, where a > b > 0, be a hyperbola in the xy-plane whose conjugate axis LM

subtends an angle of 60º at one of its vertices N. Let the area of the triangle LMN be 34 .

List-I List-II(P) The length of the conjugate axis of H is (1) 8

(Q) The eccentricity of H is (2)3

4

(R) The distance between the foci of H is (3)3

2

(S) The length of the latus rectum of H is (4) 4

The correct option is:(A) P 4; Q 2; R 1; S 3 (B) P 4; Q 3; R 1; S 2(C) P 4; Q 1; R 3; S 2 (D) P 3; Q 4; R 2; S 1


SECTION-B

(JEE-MAIN Previous Year's Questions)

Q.1 A parabola has the origin as its focus and the line x = 2 as the directrix. Then the vertex of the parabolais at(1) (1, 0) (2) (0, 1) (3) (2, 0) (4) (0, 2) [AIEEE-2008]

Q.2 A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is2

1. Then the

length of the semi-major axis is -

(1)3

2(2)

3

4(3)

3

5(4)

3

8[AIEEE 2008]

Q.3 The ellipse x2 + 4y2 = 4 is inscribed in a rectangle aligned with the coordinate axes, which is turn ininscribed in another ellipse that passes through the point (4, 0). Then the equation of the ellipse is(1) x2 + 12y2 = 16 (2) 4x2 + 48y2 = 48 (3) 4x2 + 64y2 = 48 (4) x2 + 16y2 = 16.

[AIEEE 2009]

Q.4 If two tangents drawn from a point P to the parabola y2 = 4x are at right angles, then the locus of P is(1) x = 1 (2) 2x + 1 = 0 (3) x = – 1 (4) 2x – 1 = 0

[AIEEE-2010]

CONIC SECTION


Q.5 Equation of the ellipse whose axis are the axis of coordinates and which passes through the point (–3, 1)

and has eccentricity5

2is [AIEEE 2011]

(1) 3x2 + 5y2 – 32 = 0 (2) 5x2 + 3y2 – 48 = 0(3) 3x2 + 5y2 – 15 = 0 (4) 5x2 + 3y2 – 32 = 0

Q.6 Statement 1 : An equation of a common tangent to the parabola y2 = 16 3 x and

the ellipse 2x2 + y2 = 4 is y = 2x + 2 3 .

Statement 2 : If the line y = mx +m

34, (m 0) is a common tangent to the parabola y2 = 16 3 x

and the ellipse 2x2 + y2 = 4, then m satisfies m4 + 2m2 = 24.(1) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 2.(2) Statement 1 is true, Statement 2 is false.(3) Statement 1 is false, Statement 2 is true.(4) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.

[AIEEE 2012]

Q.7 An ellipse is drawn by taking a diameter of the circle (x – 1)2 + y2 = 1 as its semi-minor axis and adiameter of the circle x2 + (y – 2)2 = 4 as its semi-major axis. If the centre of the ellipse is at the originand its axes are the coordinate axes, then the equation of the ellipse is(1) 4x2 + y2 = 8 (2) x2 + 4y2 = 16 (3) 4x2 + y2 = 4 (4) x2 + 4y2 = 8

[AIEEE 2012]

Q.8 Given : A circle 2x2 + 2y2 = 5 and a parabola y2 = 4 5 x.

Statement-I: An equation of a common tangent to these curves is y = x + 5 .

Statement-II: If the line y = mx +m

5(m 0) is their common tangent, then m satisfies

m4 – 3m2 + 2 = 0.(1) Statement-I is true, Statement-II is true, Statement-II is not a correct explanation for Statement-I.(2) Statement-I is true, Statement-II is false.(3) Statement-I is false, Statement-II is true.(4) Statement-I is true, Statement-II is true, Statement-II is a correct explanation for Statement-I.

[JEE Main 2013]

Q.9 The equation of the circle passing through the foci of the ellipse9

y

16

x 22

= 1 and having centre at (0, 3)

is(1) x2 + y2 – 6y + 7 = 0 (2) x2 + y2 – 6y – 5 = 0(3) x2 + y2 – 6y + 5 = 0 (4) x2 + y2 – 6y – 7 = 0 [JEE Main 2013]

CONIC SECTION


Q.10 The slope of the line touching both the parabolas y2 = 4x and x2 = – 32y is

(1)3

2(2)

2

1(3)

2

3(4)

8

1[JEE Main 2014]

Q.11 The locus of the foot of perpendicular drawn from the centre of the ellipse x2 + 3y2 = 6 on any tangentto it is(1) (x2 + y2)2 = 6x2 – 2y2 (2) (x2 – y2)2 = 6x2 + 2y2

(3) (x2 – y2)2 = 6x2 – 2y2 (4) (x2 + y2)2 = 6x2 + 2y2 [JEE Main 2014]

Q.12 Let O be the vertex and Q be any point on the parabola, x2 = 8y. If the point Pdivides the line segmentOQ internally in the ratio 1 : 3, then the locus of P is(1) y2 = 2x (2) x2 = 2y (3) x2 = y (4) y2 = x

[JEE Main 2015]

Q.13 The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to

the ellipse 15

y

9

x 22

is

(1)2

27(2) 27 (3)

4

27(4) 18 [JEE Main 2015]

Q.14 The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of itsconjugate axis is equal to half of the distance between its foci, is

(1) 3 (2)3

4(3)

3

4(4)

3

2

[JEE Main 2016]

Q.15 Let P be the point on the parabola, y2 = 8x which is at a minimum distance from the centre C of the circle,x2 + (y + 6)2 = 1. Then the equation of the circle, passing through C and having its centre at P is(1) x2 + y2 – 4x + 9y + 18 = 0 (2) x2 + y2 – 4x + 8y + 12 = 0

(3) x2 + y2 – x + 4y – 12 = 0 (4) x2 + y2 –4

x+ 2y – 24 = 0 [JEE Main 2016]

Q.16 The centres of those circles which touch the circle, x2 + y2 – 8x – 8y – 4 = 0, externallyand also touchthe x-axis, lie on(1) a parabola. (2) a circle.(3) an ellipse which is not a circle. (4) a hyperbola. [JEE Main 2016]

Q.17 The eccentricityof an ellipse whose centre is at the origin is2

1. If one of its directrices is x = – 4, then the

equation of the normal to it at

2

3,1 is

(1) 2y – x = 2 (2) 4x – 2y = 1 (3) 4x + 2y = 7 (4) x + 2y = 4[JEE Main 2017]

CONIC SECTION


Q.18 A hyperbola passes through the point P )3,2( and has foci at (± 2, 0). Then the tangent to this

hyperbola at P also passes through the point

(1) )32,23( (2) )33,22( (3) )2,3( (4) )3,2(

[JEE Main 2017]

Q.19 Tangents are drawn to the hyperbola 4x2 – y2 = 36 at the points P and Q. If these tangents intersect atthe point T(0, 3) then the area (in sq. units) ofPTQ is :

(1) 360 (2) 536 (3) 545 (4) 354

[JEE (Main) 2018]

Q.20 Two setsAand B are as under:A = {(a, b) R × R : |a – 5| < 1 and |b – 5 | < 1};B = {(a, b) R × R: 4 (a – 6)2 + 9(b – 5)2 < 36}. Then:(1) A B = (an empty set) (2) neither AB nor BA(3) B A (4) A B

[JEE (Main) 2018]

Q.21 Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis of theparabola at A and B, respectively. If C is the centre of the circle through the points P, A and B andCPB = then a value of tan is:

(1) 3 (2)3

4(3)

2

1(4) 2

[JEE (Main) 2018]

CONIC SECTION


EXERCISE-4

(Potential Problems Based on CBSE)

Q.1 Find equation of parabola if focus at (10, 0) the directrix x = –10.

Q.2 Find equation of parabola if focus at (0, 5), the directrix y = –5.

Q.3 Find equation of parabola if focus at (–3, 0), the directrix x + 5 = 0.

Q.4 Find equation of parabola if focus at (2, –3), the directrix x + 5 = 0.

Q.5 Find equation of parabola if focus at (1, 1), the directrix x – y = 3.

Q.6 Find the eccentricityof the ellipse of which the major axis is double the minor axis.

Q.7 If the minor axis of an ellipse is equal to the distance between its foci, prove that its eccentricity is2

1.

Q.8 Find the latus rectum and eccentricity of the ellipse whose semi-axes are 5 and 4.

Q.9 Find the eccentricityof the ellipse whose latus rectum is (i) half its major axis, (ii) half its minor axis.

Q.10 If the eccentricity is zero, prove that the ellipse becomes a circle.

Q.11 Find the equation to the hyperbola, referred to its axes as the axes of co-ordinates, whose transverseand conjugate axes are in length respectively2 and 3.

Q.12 Find the equation of the hyperbola, referred to its axes as the axes of co-ordinates, whose conjugate axisis 5 and which passes through the point (1, –2). Find also the equation of the conjugate hyperbola.

Q.13 Find the equation of the hyperbola whose eccentricity is 2, whose focus is (2, 0) and whose directrix isx – y = 0.

Q.14 Find the equation of the hyperbola, referred to its axes as the axes of co-ordinates, whose foci are (2, 0)

and (–2, 0) and eccentricity equal to2

3.

Q.15 Find the eccentricity of the hyperbola whose equation is 2x2 – 3y2 = 15.

CONIC SECTION


EXERCISE-5 (Rank Booster)

Q.1 From the point P(h, k) three normals are drawn to the parabola x2 = 8y and m1, m2 and m3 are theslopes of three normals

(a) Find the algebaric sum of the slopes of these three normals.

(b) If twoof thethreenormalsareatrightanglesthenthelocusofpointPisaconic, find thelatusrectumofconic.

(c) If the two normals from P are such that theymake complementaryangles with the axis then the locus ofpoint P is a conic, find a directrix of conic.

Q.2 TP and TQ are two tangents to the parabola y2 = 8x such that the normals at the points P and Q intersectthe curve at R.

(a) Find the locus of the middle points of the chord PQ.

(b) The chord PQ always passes through a fixed point F. Find the distance of F from the focus of the givenparabola.

(c) Find the locus of the centre of the circle circumscribing the triangle TPQ.

Q.3 Two equal parabolas P1 and P2 have their vertices at V1(0, 4) and V2(6, 0) respectively. P1 and P2 aretangent to each other and have vertical axes of symmetry.

(a) Find the sum of the abscissa and ordinate of their point of contact.

(b) Find the length of latus rectum.

(c) Find the area of the region enclosed by P1, P2 and the x-axis.

Q.4 An ellipse x2 + 4y2 = 4 is rotated anticlockwise through a right angle in its own plane about its centre. Ifthe locus of the point of intersection of a tangent to ellipse in its origin position with the tangent at thesame point of the ellipse in in its new position is given by the curve

(x2 + y2)2 = (x2 + y2) + µ x y where and µ are positive integers.Find the value of (+ µ).

Q.5 RectangleABCD has area 200.An ellipse with area 200passes through Aand C and has foci at B andD. Find the perimeter of the rectangle.

Q.6 Consider the parabola y2 = 4x and the ellipse 2x2 + y2 = 6, intersecting at P and Q.

(a) Prove that the two curves are orthogonal.

(b) Find the area enclosed by the parabola and the common chord of the ellipse and parabola.

(c) If tangent and normal at the pointP on the ellipse intersect the x-axis atT and G respectivelythen find thearea of the triangle PTG.

CONIC SECTION


Q.7 The normal to the hyperbola 1b

y

a

x2

2

2

2

drawn at an extremity of its latus rectum is parallel to an

asymptote. Show that the eccentricity is equal to the square root of ( ) /1 5 2 .

Q.8 A line through the origin meets the circle x2 +y2 = a2 at P & the hyperbola x2y2 = a2 at Q. If the locusof the point of intersection of the tangent at P to the circle and the tangent at Q to the hyperbola is thecurve a4(x2 a2) + x2 y4 = 0, then find the value of .

Q.9 An ellipse and a hyperbola have their principal axes along the coordinate axes and have a common foci

separated by a distance 132 , the difference of their focal semi axes is equal to 4. If the ratio of theireccentricities is 3/7. Find the equation of these curves.

Q.10 From the centre C of the hyperbola x2 – y2 = 9, CM is drawn perpendicular to the tangent at any pointof the curve, meeting the tangent at M and the curve at N. Find the value of the product (CM)(CN).

CONIC SECTION


EXERCISE-1


Q.1 B Q.2 A Q.3 D Q.4 A Q.5 D

Q.6 D Q.7 A Q.8 C Q.9 B Q.10 D

Q.11 B Q.12 A Q.13 B Q.14 C

Q.15 ABCD Q.16 BCD Q.17 (A) S, (B) Q, (C) S, (D) P, (E) S Q.18 34


Q.1 C Q.2 C Q.3 B Q.4 A Q.5 D

Q.6 B Q.7 A Q.8 C Q.9 D Q.10 B

Q.11 C Q.12 AB Q.13 ABCD Q.14 BCD Q.15 CD

Q.16 ABC Q.17 152


Q.1 B Q.2 B Q.3 C Q.4 B Q.5 B

Q.6 D Q.7 B Q.8 B Q.9 D Q.10 A

Q.11 A Q.12 BCD Q.13 ABCD Q.14 ABC Q.15 245

Q.16 125 Q.17 8 Q.18 32


Q.1 B Q.2 C Q.3 B Q.4 A Q.5 A

Q.6 A Q.7 D Q.8 C Q.9 C Q.10 A

Q.11 B Q.12 C Q.13 D Q.14 (A) Q; (B) R; (C) S; (D) P

Q.15 0011 Q.16 (a) x2 + y2 17x 6y = 0 ; (b) (26/3, 0)


Q.1 B Q.2 B Q.3 C Q.4 C Q.5 C

Q.6 A Q.7 C Q.8 B Q.9 D Q.10 ABD

Q.11 AB Q.12 (A) S; (B) R ; (C) P ; (D) Q Q.13 65 Q.14 0016

Q.15 8


Q.1 A Q.2 B Q.3 A Q.4 B Q.5 A

Q.6 C Q.7 A Q.8 A Q.9 A Q.10 B

Q.11 A Q.12 B Q.13 C Q.14 D

Q.15 (A) Q; (B) S; (C) P; (D) R Q.16 2

CONIC SECTION



Q.1 (a) (–4, – 1); (b) 5/4; (c) (1, – 1), (– 9, – 1); (d) 5x + 4 = 0, 5x + 36 = 0, (e) 9/2;

(f) (x + 4)2 + (y + 1)2 = 16; (g) (x + 4)2 + (y + 1)2 = 7

Q.2 A Q.3 D Q.4 B Q.5 B Q.6 C

Q.7 B Q.8 ABCD Q.9 ABCD Q.10 AD Q.11 A

Q.12 A Q.13 ACD Q.14 ABCD Q.15 ACD Q.16 ABD

Q.17 CD Q.18 (A) R, S; (B) Q; (C) R; (D) P

Q.19 (A) R; (B) S; (C) P; (D) Q Q.20 (A) S ; (B) P ; (C) S ; (D) P ; (E) Q


Q.1 A Q.2 B Q.3 A Q.4 B Q.5 B

Q.6 D Q.7 D Q.8 C Q.9 A Q.10 B

Q.11 B Q.12 A Q.13 D Q.14 C Q.15 D

Q.16 ABCD Q.17 AC Q.18 (A) R (B) S (C) T (D) P

Q.19 (A) S; (B) R; (C) P; (D) Q

EXERCISE-2

SECTION-A [PARABOLA]

Q.1 (a, 0) ; a Q.2 2x y + 2 = 0, (1, 4) ; x + 2y + 16 = 0, (16, 16)

Q.3 3x 2y + 4 = 0; x y + 3 = 0 Q.4 y = 4x + 72, y = 3x 33

Q.5 (4 , 0) ; y2 = 2a(x – 4a) Q.7 x2 + y2 + 18 x 28 y + 27 = 0

Q.8 25/2 Q.9 32

Q.10 (a) 4, (b) (2, – 1), (c) 8 sq. units Q.12 72

Q.14 32

SECTION-B [ELLIPSE]

Q.1 (a) 20x2 + 45y2 40x 180y 700 = 0; (b) 3x2 + 5y2 = 32

Q.4 x + y 5 = 0, x + y + 5 = 0 Q.5 16

Q.6 24 sq.units Q.7 (x 1)2 + y2 =11

3Q.9 54

Q.10 85 Q.12 bx + a 3 y = 2ab Q.13 60

Q.14 60 Q.15 (a)5

3; (b) 240 3 ; (c) 36

CONIC SECTION


SECTION-C [HYPERBOLA]

Q.1 7 x2 + 12xy 2 y2 2x + 4y 7 = 0 ; 48

5Q.2 a2 = 25/2 ; b2 = 16

Q.4 (1, 2) ; (4, 2) & (6, 2) ; 5x 4 = 0 & 5x + 14 = 0;3

32; 6 ; 8 ; y 2 = 0; x + 1 = 0 ;

4x 3y + 10 = 0 ; 4x + 3y 2 = 0.

Q.5 x + y ± 3 3 = 0 Q.6 3x + 2y 5 = 0 ; 3x 2y + 5 = 0

Q.10224

2

4

2

ba

1

b

y

a

x

Q.11

b

ba 22

Q.12 45

Q.13 y x 5

12

3

4; x 3 = 0 ; 8 sq. unit

EXERCISE-3

SECTION-A

PARABOLA

Q.1 A, D Q.2 C, D Q.3 (a) C, (b) ABD, (c) 2 Q.4 4

Q.5 D Q.6 B Q.7 A Q.8 D Q.9 D

Q.10 B Q.11 2 Q.12 4 Q.13 A, D Q.14 ABC

Q.15 ACD Q.16 A

ELLIPSE

Q.1 B, C Q.2 (i) D, (ii) C, (iii) B, C Q.3 (i) D; (ii) C; (iii)A

Q.4 C Q.5 9 Q.6 A Q.7 4 Q.8 AB

Q.9 A Q.10 C Q.11 AC

HYPERBOLA

Q.1 (i) B; (ii) B

Q.2 (i) D, (ii) A, B, (iii) (A) p, (B) s, t; (C) r ; (D) q, s Q.3 (a) (i) B; (ii) A; (b) 2

Q.4 (a) B, (b) BD Q.5 B Q.6 ABD Q.7 BCD Q.8 D

Q.9 A Q.10 C Q.11 B

SECTION-B

Q.1 1 Q.2 4 Q.3 1 Q.4 3 Q.5 1,2

Q.6 4 Q.7 2 Q.8 1 Q.9 4 Q.10 2

Q.11 4 Q.12 2 Q.13 2 Q.14 4 Q.15 2

Q.16 1 Q.17 2 Q.18 2 Q.19 3 Q.20 4

Q.21 4

CONIC SECTION


EXERCISE-4

Q.1 Q.2 x2 = 20 y Q.3 y2 = 4(x + 4)

Q.4 y2 – 14x + 6y – 12 = 0 Q.5 x2 + 2xy + y2 + 2x – 10y – 5 = 0

Q.62

3Q.8 Latus rectum =

5

32, e =

9

3Q.9 (i)

2

1(ii)

2

13

Q.11 9x2 – 4y2 – 9 = 0 Q.12 41x2 – 4y2 – 25 = 0; 41x2 – 4y2 + 25 = 0

Q.13 x2 – 4xy + y2 + 4x – 4 = 0 Q.14 (i) 45x2 – 36y2 – 80 = 0 Q.153

5

EXERCISE-5

Q.1 (a)h

4k ; (b) 2; (c) 2y – 3 = 0 Q.2 (a) y2 = 4(x + 4); (b) 6, (c) y2 – x + 4 = 0

Q.3 (a) 5, (b) 9/2, (c) 2234 Q.4 11

Q.5 80 Q.6 (b) 8/3, (c) 4

Q.8 4 Q.9 136

y

49

x 22

; 14

y

9

x 22

Q.10 9

in Last Nineteen Years

bcpldtpbc 201701 jee 17-18

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