chapter wise formula book of mathematics part-2 (1)

8/10/2019 Chapter Wise Formula Book of Mathematics Part-2 (1)

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(vi) cos2A + cos2B + cos2C = 1 2cosA cosB cosC

(vii) tanA + tanB + tanC = tanA tanB tanC

(viii) cotB cotC + cotC cotA + cotA cotB = 1

(ix) tan A/2 tan B/2 = 1 (x) cot A cot B =1

(xi) cot A/2 = cot A/2

11. Some useful series:

(i) sin+ sin (+ ) + sin(+ 2) + + to nterms =

2sin

2

nsin

2

1nsin

, 2 n

(ii) cos+ cos(+ ) + cos(+ 2) + . + to nterms =

2sin

2

nsin

2

1ncos

2 n

(iii) cos. Cos2. Cos22. Cos(2n1) =

sin2

2sinn

n

, n

= 1, = 2 k= 1, = (2k + 1)

Trigonometric Equation

1. General solution of the equations of the form

(i) sin= 0 = n, n I (ii) cos= 0 = (2n +1) /2, n I

(iii) tan= 0 = n, n I (iv) sin= 1 = 2n+2

, n I

(v) sin= 1 = 2n,

(iv) sin= 1 = n2

or 2n+

2

3

(vii) cos= 1 = (2n +1), (viii) sin= sin = n+(1)n(ix) cos= tan = n+ (x) tan = tan = n+

(xi) sin2= sin2 = n (xii) cos2= cos2 = n

(xiii) tan2= tan2 = n

2. For general solution of the equation of the form:

acos+ bsin= c, where c 22 ba , divide both side by 22 ba and put22 ba

a

cos,

22 ba

b

Thus the equation reduces to form cos ( ) =22 ba

c

= cos(say)

now solve using above formula.3. Some important points:

(i) If while solving an equation, we have to square it, then the roots found after squaring must be checked wheather

they satisfy the original equation or not.

(ii) If two equations are given then find the common values of between 0 & 2and then add 2nto

this common solution (value).

Inverse Trigonometric Function

1. If y = sin x, then x = sin1y, similarly for other inverse T- functions

2. Domain and Range of Inverse T- Function:Function Domain (D) Range (R)


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sin1x 1 x 1 22

cos1x 1 x 1 0

tan1x < x < 22

cot1x < x < 0

sec1

x x 1, x 1 0 , 2

cosec 1x x 1, x 1 0,22

3. Properties of Inverse T- Functions:

(i) sin1(sin ) = provided 2

2

cos1(cos ) = provided

tan1(tan ) = provided 2

< 0, xy < 1

(ii) tan1x + tan1y = + tan1

xy1

yx; if x > 0, y > 0, xy > 1


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18

(iii) tan1x tan1y = tan1

xy1

yx; if xy > 1

(iv) tan1x tan1y = + tan1

xy1

yx; if x > 0, y < 0, xy < 1

(v) tan1x + tan1y + tan1z = tan1

zxyzxy1

xyzzyx

(vi) sin1x sin1y = sin1

22 x1yy1x ; if x,y 0 & x2+ y21

(vii) sin1x sin1y = sin1

22 x1yy1x ; if x,y 0 & x2+ y2> 1

(viii) cos1x cos1y = cos1

22 y1x1xy ; if x,y > 0 & x2+ y21

(ix) cos1x cos1y = cos1

22 y1x1xy ; if x,y > 0 & x2+ y2> 1

6. Inverse trigonometric ratios of multiple angles

(i) 2sin1x = sin1(2x 2x1 ), if 1 x 1 (ii) 2cos1x = cos1(2x21), if 1 x 1

(iii) 2tan1x = tan1

2x1

x2= sin1

2x1

x2= cos1

2

2

x1

x1

(iv) 3sin1x = sin1(3x 4x3) (v) 3cos1x = cos1(4x3 3x)

(vi) 3tan1x = tan1

2

3

x31

xx3

PROPERTIES & SOLUTION OF TRIANGLE

Properties of triangle:

1. A triangle has three sides and three angles. In any ABC, we write BC = a, AB = c, AC = bA

CB

c bA

CB

a

and BAC = A, ABC = B, ACB = C

2. In ABC :(i) A + B + C = (ii) a + b > c, b + c > a, c + a > b

(iii) a > 0, b > 0, c > 0

3. Sine formula:

Csin

c

Bsin

b

Asin

a = k (say) or

c

Csin

b

Bsin

a

Asin = k (say)

4. Cosine formula:

cos A =bc2

acb 222 cos B =

ac2

bac 222 cos C =

ab2

cba 222

5. Projection formula:

A = b cos C + c cos B b = c cos A + a cos C c = a cos B + b cos A

6. Napiers Analogies:


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19

tan2

Ccot

ba

ba

2

BA

tan2

Acot

cb

cb

2

CB

tan2

Bcot

ac

ac

2

AC

7. Half angled formula In any ABC:

(a) sinbc

)cs)(bs(

2

A sin

ca

)as)(cs(

2

B

sinbc

)bs)(as(

2

C where 2s = a + b + c

(b) cosbc

)as(s

2

A cos

ca

)bs(s

2

B cos

ab

)cs(s

2

C

(c) tan)as(s

)cs)(bs(

2

A

tan)bs(s

)as)(cs(

2

B

tan)cs(s

)as)(bs(

2

C

8. ,Area of triangle :

(i) =2

1ab sin C =

2

1bc sin A =

2

1ca sin B (ii) = )cs()bs()as(s

9. tans

cs

2

Btan

2

A tan

s

as

2

Ctan

2

B tan

s

bs

2

Atan

2

C

10. Circumcircle of triangle and its radius :

(i) R =Csin2

c

Bsin2

b

Asin2

a (ii) R =

4abc

Where R is circumradius

11. Incircle of a triangle and its radius :

(iii) r =s

(iv) r = (s a) tan

2

A= (s b) tan

2

B= (s c) tan

2

C

(v) r = 4R sin2

Asin

2

Bsin

2

C (vi) cos A + cos B + cos C = 1 +

R

r

(vii) r =

2

Ccos

2

Asin

2

Bsinc

2

Bcos

2

Csin

2

Asinb

2

Acos

2

Csin

2

Bsina

12. The radii of the escribed circles are given by :

(i) r1=as

, r2=

bs

, r3=cs

(ii) r1= s tan

2

A, r2= s tan

2

B, r3= s tan

2

C

(iii) r1= 4R sin2

Acos

2

Bcos

2

C, r2= 4R cos

2

Asin

2

Bcos

2

C, r3= 4R cos

2

Acos

2

Bsin

2

C

(iv) r1+ r2+ r3 r = 4R (v)r

1

r

1

r

1

r

1

321

(vi)2

222

22

3

2

2

2

1

cba

r

1

r

1

r

1

r

1

(vii)

Rr2

1

ab

1

ca

1

bc

1

(viii) r1r2+ r2r3+ r3r1= s2 (ix) = 2R2sin A sin B sin C = 4Rr cos

2

Acos

2

Bcos

2

C

(x) r1 =

2

Ccos

2

Bcos

2

Acosc

r,

2

Bcos

2

Acos

2

Ccosb

r,

2

Acos

2

Ccos

2

Bcosa

32

HEIGHT AND DISTANCE

1. Angle of elevation and depression :

If an observer is at O and object is at P than XOP is called angle of elevation of P as seen from O.


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Horizontal line

Angle of depration

Angle of elevation

Horizontal lineX

PQ

O

If an observer is at P and object is at O, then QPO is called angle of depression of O as seen from P.

2. Some useful result:

(i) In any triangle ABC if AD : DB = m : n ACD = , BCD = & BDC = C

BA

c b

m n

BB

D

then (m + n) cot= m cot= ncot = ncotA mcotB [m n Theorem]

(ii) d = h (cot cot )

h

d

POINT

1. Distance formula:

Distance between two points P(x1, y1) and Q(x2, y2) is given by d(P, Q) = PQ = 212212 )yy()xx(

= 22 )coordinateyofDifference()coordinatexofDifference(

Note:(i) d(p, Q) 0 (ii) d(P,Q) = 0 P = Q

(iii) d(P, Q) = d(Q, P) (iv) Distance of a point (x, y) from origin (0, 0) = 22 yx

2. Use of Distance Formula :

(a) In Triangle:Calculate AB, BC, CA

(i) If AB = BC = CA, then is equilateral. (ii) If any two sides are equal then is isosceles.

(iii) If sum of square of any two sides is equal to the third, then is right triangle.

(iv) Sum of any two equal to left third they do not form a triangle i.e. AB = BC + CA or BC = AC + AB or AC = AB+ BC. Here points are collinear.

(b) In Parallelogram :

Calculate AB, BC, CD and AD.

(i) If AB = CD, AD = BC, then ABCD is a parallelogram.

(ii) If AB = CD, AD = BC and AC = BD, then ABCD is a rectangle.

(iii) If AB = BC = CD = AD, then ABCD is a rhombus.

(iv) If AB = BC = CD = AD and AC = BD, then ABCD

(c) For circumcentre of a triangle :

Circumcentre of a triangle is equidistant from vertices i.e. PA = PB = PC.

Here P is circumcentre and PA is radius.(i) Circumcentre of an acute angled triangle is inside the triangle.

(ii) Circumcentre of a right triangle is mid point of the hypotenuse.


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x y1 1

x y2 2

x y3 3

x y1 1

x y1 1

......

x y2 2x y

3 3

x yn nx y1 1

(iii) Circumcentre of an obtuse angled triangle is outside the triangle.

3. Section formula:

(i) Internally :

n

m

BP

AP = , Here > 0

m n

PA (x ,y )1 1 B (x ,y )1 1 P

nm

nymy,

nm

nxmx 1212

(ii) Externally :

n

m

BP

AP = ,

PA (x ,y )1 1 B (x ,y )1 1

m

n

P

nm

nymy,

nm

nxmx 1212

(iii) Coordinates of mid point of PQ are

2

yy,

2

xx 2121

(iv) The line ax + by + c = 0 divides the line joining the points

(x1,y1) & (x2, y2) in the ratio = )cbyax(

)cbyax(

22

1

(v) For parallelogram midpoint diagonal AC = mid point of diagonal BD

(vi) Coordinates of centroid G

3

yyy,

3

xxx 321321

(vii) Coordinates of incentre I

cba

cybyay,

cba

cxbxax 321321

(viii) Coordinates of orthocenter are obtained by solving the equation of any two altitudes.

4. Area of Triangle:

The area of triangle ABC with vertices A(x1, y1), B (x2, y2) and C(x3, y3).

=1yx

1yx 1yx

2

1

33

2211

(Determinant method)

=2

1 =

2

1[x1y2+ x2y3+ x3y1 x2y1 x3y2 x1y3]

[Stair Method]

Note:

(i) Three points A, B, C are collinear if area of triangle is zero.

(ii) If in a triangle point arrange in anticlockwise then value of be +ve and if in clockwise then will be ve.

5. Area of Polygon:

Area of polygon having vertices (x1, y1), (x2, y2), x3, y3) .(xn, yn) is given by area

=2

1 . Points must be taken in order.

6. Rotational Transformation:

If coordinates of any point P(x, y) with reference to new axis will be (x, y) then


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cossin'y

sincos'x

yx

7. Some important points:

(i) Three pts. A, B, C are collinear, if area of triangle is zero

(ii) Centroid G of ABC divides the median AD or BE or CF in the ratio 2 : 1

(iii) In an equilateral triangle, orthocenter, centroid, circumcentre, incentre coincide.

(iv) Orthocentre, centroid and circumcentre are always collinear and centroid divides the line joining orthocenter and

circumcentre in the ratio 2 : 1

(v) Area of triangle formed by coordinate axes & the line ax + by + c = 0 isab2

c2.

STRAIGHT LINE

1. Slope of a line:

The tangent of the angle that a line makes with +ve direction of the x-axis in the anticlockwise sense is called slope or

gradient of the line and is generally denoted by m. Thus m = tan .

(i) Slope of line || to x-axis is m = 0

(ii) Slope of line || to y-axis is m = (not defined)(iii) Slope of the line equally inclined with the axes is 1 or 1

(iv) Slope of the line through the points A(x1, y1) and B(x2, y2) is12

12

xx

yy

.

(v) Slope of the line ax + by + c = 0, b 0 is b

a

(vi) Slope of two parallel lines are equal.

(vii) If m1& m2are slopes of two lines then m1m2= 1.

2. Standard form of the equation of a line :

(i) Equation of x-axis is y = 0 (ii) Equation of y-axis is x = 0

(iii) Equation of a straight line || to x-axis at a distance b from it is y = b

(iv) Equation of a straight line || to y-axis at a distance a from it is x = a

(v) Slope form :Equation of a line through the origin and having slope m is y = mx.

(vi) Slope Intercept form :Equation of a line with slope m and making an intercept c on the y-axis is y = mx +

c.

(vii) Point slope form :Equation of a line with slope m and passing through the point (x1, y1) is y y1=

m(x x1)

(viii)Two points form :Equation of a line passing through the points (x1, y1)&(x2, y2)is12

1

12

1

xx

xx

yy

yy

(ix) Intercept form:Equation of a line making intercepts a and b respectively on x-axis and y-axis is .1b

y

a

x

(x) Parametric or distance or symmetrical form of the line:Equation of a line passing through (x1, y1) and

making an angle , 0 , 2

with the +ve direction of x-axis is

sin

yy

cos

xx 11 = r

x = x1+ r cos , y = y1+ r sin where r is the distance of any points P(x, y) on the line from the point (x1, y1)

(xi) Normal or perpendicular form :Equation of a line such that the length of the perpendicular from the origin on it

is p and the angle which the perpendicular makes with the +ve direction of x-axis is , is x cos + y sin = p.

3. Angle between two lines:

(i) Two lines a1x + b1y + c1= 0 & a2x + b2y + c2= 0 are

(a) Parallel if 2

1

2

1

2

1

c

c

b

b

a

a

(b) Perpendicular if a1a2+ b1b2= 0


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(c) Identical or coincident if2

1

2

1

2

1

c

c

b

b

a

a (d) If not above three, then = tan1

2121

21l2

bbaa

baba

(ii) Two lines y = m1x + c and y = m2x + c are

(a) Parallel if m1= m2 (b) Perpendicular if m1m2= 1

(c) If not above two, then = tan121

21

mm1

mm

4. Position of a point with respect to a straight line :The line L(xi, yi) i = 1, 2 will be of same sign or of opposite sign according to the point A(x1, y1) & B (x2, y2) lie on

same side or on opposite side of L (x, y) respectively.

5. Equation of a line parallel (or perpendicular) to the lineax + by + c = 0 is ax + by + c = 0 (or bx ay + = 0)

6. Equation of st. lines through (x1,y1) making an angle with y = mx + c isy y1=

tanm1

tanm

(x x1)

7. length of perpendicular from (x1, y1) on ax + by + c = 0 is22

11

ba

cbyax|

8. Distance between two parallel lines ax + by + ci= 0, i =1,2 is22

21

ba

|cc|

9. Condition of concurrency for three straight lines Li aix + biy + ci= 0, i = 1, 2, 3 is

333

222

111

cba

cba

cba

= 0

10. Equation of bisectors of angles between two lines:

22

22

222

21

21

111

ba

cybxa

ba

cybxa

11. Family of straight lines :

The general equation of family of straight line will be written in one parameterThe equation of straight line which passes through point of intersection of two given lines L 1and L2 can be taken as

L1+ L

2= 0

12. Homogenous equation:If y = m1x and y = m2x be the two equations represented by

ax2+ 2hxy + by2=, 0 then m1+ m2= 2h/b and m1m2= a/b

13. General equation of second degree:

ax2+ 2hxy + by2+ 2gx + 2fy + c = 0 represents a pair of straight line if =cfg

fbh

gha

= 0

If y = m1x + c & y = m2x + c represents two straight lines then m1+ m2= ,b

h2m1m2=

b

a.

14. Angle between pair of straight lines:

The angle between the lines represented by ax2+ 2hxy + by2+ 2gx + 2fy + c = 0 or ax2+ 2hxy + by2= 0 is tan =

)ba(

abh2 2

(i) The two lines given by ax2+ 2hxy + by2= 0 are

(a) Parallel and coincident iff h2 ab = 0 (b) Perpendicular iff a + b = 0(ii) The two line given by ax2+ 2hxy + by2+ 2gx + 2fy + c = 0 are

(a) Parallel if h2 ab = 0 & af2= bg2(b) Perpendicular iff a + b = 0(c) Coincident iff g2 ac = 0

13. Combined equation of angle bisector of the angle between the lines ax2+ 2hxy + by2= 0

h

xy

ba

yx 22

Circle

1. General equation of a circle:x2+ y2+ 2gx + 2fy + c = 0 where g, f and c are constants


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(i) Centre of the circle is (g, f)

yof.coeff

2

1,xof.coeff

2

1e.i

(ii) Radius is cfg 22

2. Central (Centre radius) form of a circle:

(i) (x h)2+ (y k)2= r2, where (h, k)is circle centre and r is the radius

(ii) x2+ y2= r2, where (0 ,0) origin is circle centre and r is the radius

3. Diameter form :If (x1,y1) and (x2, y2) are end pts. Of a diameter of a circle, then its equation is(x x1) (x x2) + (y y1) (y y2) = 0

4. Parametric equation :

(i) The parametric equation of the circle x2+ y2= r2are x = r cos , y = rsin,

where point (r cos, r sin )

(ii) The parametric equations of the circle (x h)2+ (y k)2= r2are x = h + rcos, y = k + rsin

(iii) The parametric equation of the circle x2+y2+ 2gx + 2fy + c = 0 are x = g + coscfg 22 ,

y = f + sincfg 22

(iv) For circle x2+ y2= a2= a2, equation of chord joining 1&

2is

2cosx 21

+ y

2sin 21

= r2

cos 21

5. Concentric circles:Two circles having same centre C (h,k) but different radii r1& r2respectively are called concentric

circles.

6. Position of a point w.r.t. a circle:A point (x1,y1) lies outside, on or inside a circle

S x2+ y2+ 2gx + 2fy + c = 0 according as

S1 cfy2gx2yx 1121

21 is +ve, zero or ve

7. Chord length (length of intercept) = 2 22 pr

8. Intercepts made on coordinate axes by the circle:

(i) x axis = 2 cg2 (ii) y axis = 2 cf2

9. Length of tangent = 1S

10. Length of the intercept made by line:y = mx + c with the circle x2+ y2= a2is

2

222

m1

c)m1(a2

or (1 + m2) | x1 x2| where | x1 x2| = difference of roots i.e.

a

D.

11. Condition of Tangency:Circle x2+ y2= a2will touch the line y = mx + c if c = a 2m1 .

12. Equation of tangent, T = 0:

(i) Equation of tangent to the circle x

2

+ y

2

+ 2gx + 2fy + c = 0 at any point (x 1, y1) isxx1+ yy1+ g(x + x1) + f( y + y1) + c = 0

(ii) Equation of tangent to the circle x2+ y2= a2at any point (x1, y1) is xx1+ yy1= a2

(iii) In slope form : From the condition of tangency for every value of m.

The line y = mx a 2m1 is a tangent to the circle x2 + y2= a2and its point of contact is

22 m1

a,

m1

am

(iv) Equation of tangent at (a cos , a sin ) to the circle x2+ y2= a2is x cos + y sin = a.

13. Equation of normal :

(i) Equation of normal to the circle x2

+ y2

+ 2gx + 2fy + c = 0 at any point P(x 1, y1) isy y1=

gx

fy

1

1

(x x1)


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(ii) Equation of normal to the circle x2+ y2= a2at any point (x1,y1) is xy1 x1y = 0

14. Equation of pair of tangents SS1= T2:

15. The point of intersection of tangents drawn to the circle x2+ y2= r2at point 1& 2is given as

2cos

2cosr

,

2cos

2cosr

21

21

21

21

16. Equation of the chord of contact of the tangents drawn from point P outside the circle is T = 017. Equation of a chord whose middle pt. is given by T = S1

18. Director circle :Equation of director circle for x2+ y2= a2is x2+ y2= 2a2. Director circle is a concentric circle whose

radius is 2 times the radius of the given circle.

19. Equation of polar of point(x1,y1) w.r.t. the circle S = 0 is T = 0

20. Coordinates of pole :Coordinates of pole of the line lx + my + n = 0 w.r.t the circle x2+ y2= a2are

n

ma,

n

la 22

21. Family of circles:

(i) S + S = 0 represents a family of circles passing through the pts. of intersection of S = 0

& S = 0 if 1(ii) S + L = 0 represents a family of circles passing through the point of intersection of S = 0

& L = 0

(iii) Equation of circle which touches the given straight line L = 0 at the given point (x1, y1) is given as (x x1)2+ (y

y1)2+ L = 0.

(iv) Equation of circle passing through two points A(x1,y1) & B(x2, y2) is given as

(x x1) (x x2) + (y y1) (y y2) + .01yx

1yx

1yx

22

11

22. Equation of Common Chord is S S1= 0.

23. The angle

of intersection of two circles with centres C1& C2and radii r1& r2is given by cos=

21

221

21

rr2

drr , where d = C1C2

24. Position of two circles :Let two circle with centres C1, C2and radii r1, r2.

Then following cases arise as

(i) C1C2> r2+ r2do not intersect or one outside the other, 4 common tangents.

(ii) C1C2= r1+ r2Circles touch externally, 3 common tangents.

(iii) | r1 r2| < C1C2< r1+ r2Intersection at two real points, 2 common tangents.

(iv) C1C2= |r1 r2| internal touch, 1 common tangent.

(v) C1C2< |r1+ r2| one inside the other, no tangent.Note:Point of contact divides C1C2in the ratio r1: r2internally or externally as the case may be

25. Equation of tangent at point of contact of circle is S1 S2= 0

26. Radical axis and radical centre :

(i) Equation of radical axis is S S1= 0

(ii) The point of concurrency of the three radical axis of three circles taken in pairs is called radical centre of three

circles.

27. Orthogonality condition:

If two circles S x2+ y2+ 2gx + 2fy + c = 0 and S = x2+ y2+ 2gx + 2fy + c = 0 intersect each other orthogonally,

then 2gg + 2ff = c + c.

PARABOLA


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26

1. Standard Parabola :

Imp. Terms y2= 4ax y

2= 4ax x

2= 4ay x

2= 4ay

Vertex (v) (0, 0) (0, 0) (0, 0) (0, 0)

Focus (f) (a, 0) (a, 0) (0, a) (0, a)

Directrix (D) x = a x = a y = a y = a

Axis y = 0 y = 0 x = 0 x = 0

L.R. 4a 4a 4a 4a

Focal x + a a x y + a a y

Distance

Parametric (at2, 2at) ( at2, 2at) (2at, at2) (2at, at2)

Coordinates

Parametric x = at2 x = at2 x = 2at x = 2at

Equations y = 2at y = 2at y = 2at2 y = at2

Y

A

(a,2a)

(a,2a)

Directrix

x=

a

Tangent at the vertex

Latus Rectum

axis of the

parabola i.e. y =0

Vertex (0,0)L

S(a,0)

Focusx

y2= 4ax

L

Directrix

x=a

Latus Rectum

axis of theparabola

S

A(a,0)

(a,2a)

L(a,2a)

y2= ax

L

y = a

x

Latus Rectum

Tangent atthe vertexi.e. y =0

S

A

Focus

Y

(a,2a)L

(0,a)(a,2a)

(0,b)


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x2= 4ay

L

y = a

x

Latus Rectum

S

A

Y

(a,2a) L (a 2a)

(a 2a)

x2= 4ay

2. Special form of Parabola

Parabola which has vertex at (h, k), latus rectum and axis parallel to x-axis is (y k)2= (x h)

axis is y = k and focus at

k,

4h

Parabola which has vertex at (h, k), latus rectum and axis parallel to y-axis is (x h)2= (y k)

axis is x = h and focus at

4k,h

Equation of the form ax2+ bx + c = y represents parabola.

i.e. y 22

a2

bxa

a4

bac4

,with vertex

a4

bac4,

a2

b 2and axes parallel to y-axis

Note :Parametric equation of parabola (y k)2= 4a(x h) are x = h + at2, y = k + 2at

3. Position of a point (x1, y1) and a line w.r.t. parabola y2= 4ax.

The point (x1, y1) lies outside, on or inside the parabola y2= 4ax according as y1

2 4ax1>, = or < 0

The line y = mx + c does not intersect, touches, intersect a parabola y2= 4ax according as c

> = < a/mNote: Condition of tangency for parabola y2= 4ax, we have c = a/m and for other parabolas check disc.D = 0.

4. Equations of tangent in different forms:

(i) Point form / Parametric formEquations of tangent of all other standard parabolas at (x1, y1) / at t (parameter)

Equation of

parabolaTangent at (x1, y1)

Parametric

coordinates tTangent of t

y2= 4ax yy1= 2a(x + x1) (at2, 2at) ty = x + at2

y2= 4ax yy1= 2a(x + x1) ( at2, 2at) ty = x + at2

x2= 4ay xx1= 2a(y + y1) (2at, at2) tx = y + at2x2= 4ay xx1= 2a(y + y1) (2at, at

2) tx = y + at2

(ii) Slope formEquations of tangent of all other parabolas in slope form

Equation of

parabolas

Point of contact in

terms of slops(m)

Equations of tangent in

terms of slope (m)

Condition of

Tangency

y2= 4ax

m

a2'

m

a2

y = mx +m

a c =

m

a

y2= 4ax

m

a2

'm

a2 y = mx m

a

c = m

a

x2= 4ay (2am, am2) y = mx am2 c = am2


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x2= 4ay (2am, am2) y = mx + am2 c + am2

5. Point of intersection of tangents at any two points P(at12, 2at1) and Q(at2

2, 2at2) on the parabola y2= 4ax is (at1t2, a(t1+

t2)) i.e. (a(G.M.)2, a(2A.M.))

6. Combined equation of the pair of tangents drawn from a point to a parabola is SS = T2, where S = y2 4ax, S = y12

4ax1and T = yy1 2a(x + x1)

7. Equations of normal in different forms

(i) Point form / Parametric formEquations of normals of all other standard parabolas at (x1, y1) / at t (parameter)

Equation of

parabola

Normal at

(x1, y1)Point t

Normals

at t

y2= 4axy y =

a2

y1 (x x1)(at2, 2at) y + tx = 2at + at3

y2= 4axy y =

a2

y1 (x x1)(at2, 2at) y-tx = 2at + at3

x2= 4ayy y =

1x

a2(x x1)

(2at, at2) x + ty = 2at + at3

x2= 4ayy y =

1x

a2(x x1)

(2at, at2) x ty = 2at + at3

(ii) Slope formEquations of normal, point of contact, and condition of normality in terms of slope (m)

Equation of

parabola

Point of

contact

Equations of normal Condition of

Normality

y2= 4ax (am2, 2am) y = mx 2am am3 c = 2am am3

y2= 4ax (am2, 2am) y = mx + 2am +am3 c = am + am3

x2= 4ay

2m

a,

m

a2 y = mx + 2a + 2m

a c = 2a +

2m

a

x2= 4ay

2m

a,

m

a2 y = mx 2a 2

m

a c = 2a

2

m

a

Note:

(i) In circle normal is radius itself.

(ii) Sum of ordinates (y coordinate) of foot of normals through a point is zero.

(iii) The centroid of the triangle formed by taking the foot of normals as a vertices of concurrent normals of y2= 4ax

lies on x-axis.

8. Condition for three normals from a point (h, 0) on x-axis to parabola y2= 4ax

(i) We get 3 normals if h > 2a (ii) We get one normal if h 2a.

(iii) If point lies on x-axis, then one normal will be x-axis itself.

9. (i) If normal of y2= 4ax at t1meet the parabola again at t

2then t

2= t

1

1t

2

(ii) The normals to y2= 4ax at t1and t2intersect each other at the same parabola at t3, then t1t2= 2

and t3= t1 t2

10. (i) Equation of focal chord of parabola y2= 4ax at t1is y =1t

t221

1

(x a)

If focal chord of y2= 4ax cut (intersect) at t1and t2then t1t2= 1 (t1must not be zero)

(ii) Angle formed by focal chord at vertex of parabola is tan =3

2|t2 t1|

(iii) Intersecting point of normals at t1and t2on the parabola y2= 4ax is

(2a + a(t12+ t2

2+t1t2), at1t2(t1+ t2))

11. Equation of chord of parabola y2= 4ax which is bisected at (x1, y1) is given by T = S1


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29

12. The locus of the mid point of a system of parallel chords of a parabola is called its diameter. Its equation is y =m

a2

.

13. Equation of polar at the point (x1, y1) with respect to parabola y2= 4ax is same as chord of contact and is given by T =

0 i.e. yy1= 2a (x + x1) Coordinates of pole of the line x + my + n = 0 w.r.t. the parabola y2= 4ax is

am2,

n

14. Diameter :It is locus of mid point of set of parallel chords and equation is given by T = S1

15. Important results for Tangent :

(i) Angle made by focal radius of a point will be twice the angle made by tangent of the point with axis of parabola

(ii) The locus of foot of perpendicular drop from focus to any tangent will be tangent at vertex.

(iii) If tangents drawn at ends point of a focal chord are mutually perpendicular then their point of intersection will lie

on directrix.

(iv) Any light ray traveling parallel to axis of the parabola will pass through focus after reflection through parabola.

(v) Angle included between focal radius of a point and perpendicular from a point to directrix will be bisected of

tangent at that point also the external angle will be bisected by normal.

(vi) Intercepted portion of a tangent between the point of tangency and directrix will make right angle at focus.

(vii) Circle drawn on any focal radius as diameter will touch tangent at vertex.

(viii) Circle drawn on any focal chord as diameter will touch directrix.

ELLIPSE

1.

Standard Ellipse (e < 1)

1b

y

a

x2

2

2

2

Ellipse Imp.terms

For a > b For b > a

Centre (0,0) (0,0)

Vertices (a,0) (0, b)

Length of major axis 2a 2b

Length of minor axis 2b 2a

Foci (ae, 0) (0, be)

Equation of directrices x = a/e y b/e

Relation in a, b and c b2 = a2(1 a2) a2= b2(1 e2 )

Length of latus rectum 2b2/a 2a2/b

Ends of latus rectum

a

b,ae

2

be,

b

a2

Parametric coordinates (a cos , b sin) ( acos , b sin)

0 < 2

Focal radii SP = a ax1 SP = b ey1

SP = a + ex1 SP = b + ey1

Sum of focal radii SP + SP = 2a 2b

Distance btnfoci 2ae 2be

Distance btndirectrices 2a/e 2b/e

Tangent at the vertices x = a, x = a y = b, y = b

Note:If P is any point on ellipse and length of perpendiculars from to minor axis and major axis are p1& p2, then |xp|

= p1, |yp| = p2 222

2

21

b

p

a

p =1


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30

Directrix

Directrix

Y

Y

A AS

C

(a,0)(a,0)

MZ Z

S

B

x=a/ex= a/e

xX

(0,b)

(0,b)

(ae,0)

p(x,y)

(ae,0)

a > b

XBC

(0

,b

e)

Z

S

Z

B(a,0) (a,0)

(0,0)

y =b/c k

A(0,b)

X

A(0, b)

S

2. Special form of ellipse:

If the centre of an ellipse is at point (h, k) and the directions of the axes are parallel to the coordinate axes, then its

equation is .1b

)ky(

a

)hx(

2

2

2

2

3. Auxillary Circle:The circle described by taking centre of an ellipse as centre and major axis as a diameter is called an

auxillary circle of the ellipse. If2

2

2

2

b

y

a

x = 1 is an ellipse then its auxillary circle is x2+ y2= a2.

Note:Ellipse is locus of a point which moves in such a way that it divides the normal of a point on diameter of a point of

circle in fixed ratio.

4. Position of a point and a line w.r.t. an ellipse:

The point lies outside, on or inside the ellipse if S1= 221

2

21

b

y

a

x 1 >, = or < 0

The line y = mx + c does not intersect, touches, intersect, the ellipse if a2

m

2

+ b

2

< = > c

2

5. Equation of tangent in different forms :

(i) Point form :The equation of the tangent to the ellipse2

2

2

2

b

y

a

x = 1 at the point (x1, y1) is .1

b

yy

a

xx21

21

(ii) Slope form :If the line y = mx + c touches the ellipse2

2

2

2

b

y

a

x = 1, then c2= a2m2+ b2. Hence, the straight line y

= mx 222 bma always represents the tangents to the ellipse.

Point of contact :

Line y = mx 222 bma touches the ellipse2

2

2

2

b

y

a

x = 1 at

222

2

222

2

bma

b,

bma

ma.

(iii) Parametric form : The equation of tangent at any point (a cos , b sin ) isa

xcos +

b

ysin = 1.

chapter wise formula book of mathematics part-2 (1)

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