www.atharvlabs.com

1

Rapid Revision Sheet Coordinates and Straight Lines

By Pankaj Baluja

1. In the following, each question has four options of which one or more options are correct.

Indicate the correct option(s):

(a) P(m,n) where m,n are natural numbers, is any point in the interior of the quadrilateral

formed by the pair of lines xy 0 and the two lines 2x y 2 0 and 4x 5y 20. The

possible number of positions of the point P is

(A) six (B) four

(C) five (D) none of these (b) If 2( , ) falls inside the angle made by the lines 2y x, x 0 and y 3x, x 0 then

the set of values of is

(A) ( ,3) (B) 1

,32

(C) (0, 3) (D) 1

( ,0) ,2

(c) The image of the point A(1, 2) by the line mirror y x is the point B and the image of B

by the line mirror y 0 is the point ( , ). Then

(A) 1, 2 (B) 0, 0

(C) 2, 1 (D) None of these

(d) 2A ( 1 t t,0) and 2B ( 1 t t,2t) are two variable points where t is a

parameter. The locus of the middle point of AB is

(A) a straight line (B) a pair of lines

(C) a circle (D) None of these 2. Fill in the following incomplete statements so that the statements becomes true.

(a) The line through (0, 4) which together with the lines y (2 3)x from an equilateral

triangle, has the equation _____

(b) The set of real values of a for which the line segment AB where 2 3A a 1,a

7

and

2B (5 1,a 2) is divided into two segments by the line 2x 7y 9 is ____

(c) A square is made with the segment on the line x y

1a b cut off by the axes as a

diagonal. The vertices of the square, not on this diagonal, are ____ and ____.

(d) The intercepts made on the line x y 5 2 by the lines y x tan ; 0, , ,4 2

are in

AP. Then is ____

3. P is a point on the level ground. C is a point south-east of P at a distance 3 2 m from P. Find

a point A bearing north of P and a point B bearing west of P such that P is located at the

centroid of the ABC.

4. ABC is a triangle and D, E are the feet of the perpendiculars from A and B to the opposite

sides respectively. If D = (20, 25), E = (8, 16) and C = (10, 15) then find the orthocentre of

the triangle ABC.

www.atharvlabs.com

2

Rapid Revision Sheet Coordinates and Straight Lines

By Pankaj Baluja

5. Q is any point the line x a. If A (a,0) and the bisector QR of the angle OQA, meets the x-

axis at R, find the locus of the foot of the perpendicular from R to OQ, O being the origin.

6. Two fixed lines have the equations y 1 and y 4. The lines segment OB whose equation is

y x, is rotated about a point on it so that B reaches the lines after rotating through an

angle 45°, once anticlockwise and the other time clockwise. If B is 3 2 3 2

, ,2 2

find the

point about which OB must be rotated.

7. AB is a line joining the points (–1, 3) and (4, 1), and CD is a line whose slope is 34

and its

distance from the origin is 1. Find so that the point P(0, ) when translated parallel to the

line x y 1 passes through the point of intersection Q of the lines AB and CD. Also find the

distance through which P must be translated to reach Q.

8. Two mutually perpendicular lines are drawn through the point of intersection A of the lines

y x 0 and x y 1 to cut the line 2x y 0 at two points B and C. Find the equation of

the locus of the centroid of the triangle ABC.

9. OAB has the vertices O = (0, 0), 3

A ,02

and 3

B 0, .2

Find such that 0 2 and P (sin ,cos ) is an interior point of the OAB.

10. For the ABC, the vertex A = (4, 3), its orthocentre H = (3, 3) and the centroid G = (1, 1).

Find the equations of the sides of the ABC.

11. ABC is an isosceles triangle in which AB = AC. The images of the vertices A, B, C by the lines,

BC, CA, AB are respectively A', B' and C'. If the triangle A'B'C' is equilateral then find the angle

A.

12. Let a new distance d(P,Q) between the points 1 1P (x , y ) and 2 2Q (x , y ) be defined as

1 2 1 2d(P,Q) x x y y . Let O = (0, 0) and A = (3, 2) be two fixed points. Let R (x, y),

x 0, y 0 such that R is equidistant from the points O and A in the sense of the new

distance. Prove that the locus of R consists of a line segment of finite length and an infinite

ray. Also, sketch a labelled diagram to show the locus.

www.atharvlabs.com

3

Rapid Revision Sheet Coordinates and Straight Lines

By Pankaj Baluja

O 1 2 3

1

2

Y

X

1—2

( , 2)

5—2

x + y =

5—2

( , 0)

x =1—2

x = 3

y = 2

Answers

1. (a) A (b) B (c) C (d) C

2. (a) x y 4 (b) 1

4,2

(c) a b a b a b b a

, , ,2 2 2 2

(d) 1tan 5

3. each at 3 m from P 4. (15, 30)

5. 2 2 2 2 2 2(x y ax) y (x y ) 6. (1, 1)

7. 27 2

8,18; ,11 27

8. 12x 6y 1

9. 0 15 or 75 90 10. 2x 1 0,( 11 2)x 3y 4 11 1

11. 6

12.

www.atharvlabs.com

1

Rapid Revision Sheet Pair of Straight Lines and Transformation of Axes

By Pankaj Baluja

1. Fill in the blanks so that the resulting sentences become true.

(a) The angle between the lines given by 2 2x 2xy sec y 0 is _____

(b) The difference of the slopes of the lines 2 2 2 2 2x (sec sin ) 2xy tan y sin 0 is ___

(c) If pairs of lines 2 2x 2pxy y 0 and 2 2x 2qxy y 0 are such that each pair bisects

the angles between the other pair then the value of p q is ____

(d) If the equation 2 212x 7xy py 18x qy 6 0 represents two perpendicular lines then

p ____, q ____.

2. Choose the correct answer(s) from the given multiple options.

(a) The number of values of for which bisectors of the angle between the lines

2 2 2 2ax 2hxy by (x y ) 0 are the same is

(A) two (B) one

(C) zero (D) infinite (b) If one of the lines of 2 2ax 2hxy by 0 be the bisector of the angle between the

coordinate axes then

(A) 2 2(a b) 4h (B) 2 2(a b) 4h 0

(C) 2 2(a b) 4h (D) 2 2(a b) 4h 0

(c) The equation 2 2 2(x y 1) k(x y 1) 0 will represent two straight lines only if

(A) k 0 (B) k 3

(C) k 0 or 3 (D) None of these

(d) The equation 2 24x 24xy 11y 0 represents

(A) two parallel lines (B) a circle

(C) two perpendicular lines (D) two lines through the origin

(e) The value of c for which the lines joining the origin to the points of intersection of the line

y 3x c and the circle 2 2x y 2 are perpendicular to each other, is

(A) 1 (B) 0

(C) –2 (D) 2 3. The equations of pairs of opposite sides of a rectangle are 2x 7x 6 0 and

2y 14y 40 0. Find the equations to its diagonals.

4. Show that the orthocentre of the triangle formed by the pair of lines 2 2ax 2hxy by 0 and

the line x my 1 is given by

2 2

x y a b.

m am 2h m b

5. If the angle between the lines joining the origin to the points of intersection of the circle

2 2 2x y a and the line x cos y sin 1 be 45°, find the possible values of a where

is a constant.

www.atharvlabs.com

2

Rapid Revision Sheet Pair of Straight Lines and Transformation of Axes

By Pankaj Baluja

6. Find the equation of the pair of lines joining the point (1, 1) to the points of intersection of

the line x y 0 with the circle 2 2x y 2x 4y 1 0. Use change of the origin.

7. Prove that 2 29x 24xy 16y 45x 60y 50 0 represents two parallel lines 1L and 2L ,

and the point (2, 0) lies between the two lines. A parallelogram is constructed with two sides

on the lines 1L and 2L , and a diagonal passing through (2, 0) such that the other two sides

are perpendicular to this diagonal and the area of the parallelogram is the least possible.

Prove that one of the angles of the parallelogram will be 4 . Also, find the joint equation of

the two possible diagonals through the point (2, 0).

Answers

1. (a) or (b) 2 (c) –1 (d) 12; 232

or 1 2. (a) D (b) C (c) C (a) C, D

3. 6x 5y 14 0,6x 5y 56 0 5. 4 2 2

6. 2 23x 2xy 9y 8x 16y 4 0 7. 2 27x 48xy 7y 28x 96y 96y 28 0

www.atharvlabs.com

1

Rapid Revision Sheet Circles

By Pankaj Baluja

1. In each of the following, fill in the blanks so that the resulting statement becomes correct.

(a) The coordinates of the points on the circle 2 2x y 4 which are at a distance 5 from the

line 4x 3y 25 are ____ and _____

(b) There equation to the circle on the other side of the line x y 2 similarly situated as the

circle 2 2x y 2x 0 is _____

(c) The sum of the squares of the lengths of the chords intercepted by the lines

y x n,n N on the circle 2 2x y 100 is equal to _____

(d) The equation of the circle of radius 2 containing the point (1, –2) and touching the lines

x y is _____

2. In each of the following one or more options are correct. Choose the correct option(s).

(a) The straight line mx y 1 2m cuts the circle 2 2x y 1 at one point at least. Then the

set of values of m is

(A)

4,0

3 (B)

4 4,

3 3

(C)

40,

3 (D) None of these

(b) If (2, 5) is an interior point of the circle 2 2x y 8x 12y p 0 and the circle neither

cuts nor touches any one of the axes of coordinates then

(A) p (36, 47) (B) p (16, 47)

(C) (16, 36) (D) None the these (c) The range of values of a such that the angle between the pair of tangents drawn from

(a,0) to the circle 2 2x y 1 satisfies ,

2 is

(A) (1, 2) (B) (1, 2)

(C) ( 2, 1) (D) ( 2, 1) (1, 2)

(d) The circle of radius 1, touching the pair of lines 2 212x 25xy 12y 0, x 0 has the

equation

(A) 2 2x y 10x 10y 49 0 (B) 2 2x y 10x 10y 49 0

(C) 2 2x y 10x 10y 49 0 (D) None of these (e) The number of points (a 1, a), where a is an integer, lying inside the region bounded by

the circles 2 2x y 2x 1 0 and 2 2x y 2x 17 0, is

(A) 2 (B) 4

(C) 1 (D) None of these

3. Find the equations of the sides of a square inscribed in the circle 2 23(x y ) 4, on e of

whose sides is parallel to the line x y 7 0.

www.atharvlabs.com

2

Rapid Revision Sheet Circles

By Pankaj Baluja

4. Let A = (–1, 0), B = (3, 0) and PQ be any line passing through (4, 1). Find the range of

values of the slope of PQ for which there are two points on PQ at which AB subtends a right

angle.

5. Three concentric circles of which the biggest is 2 2x y 1, have their radii in AP. If the line

y x 1 cuts all the circles in real and distinct points then find the interval in which the

common difference of the AP will lie.

6. The equation of the common chord of two circles is x y 1. One of the circles has the ends

of a diameter at the points (1, –3) and (4, 1) and the other passes through the point (1, 2)

2 2x y 2x 0 . Find the equations of the two circles.

7. Prove that on-third of the circumference of the circle 2 2x y 2x 0 falls inside the circle

2 23(x y ) 2 3x 2y 1 0.

8. Find the locus of the centre of the circles 2 2x y 2 x 2 y 2 0 if the tangents from the

origin to each of the circles are orthogonal.

9. Let 2 2ax 2hxy by 1 be the equation of a conic section and P be a point not on the curve.

Any line through P cuts the conic section at Q and R such that PQ. PR is a constant. Show

that the conic section is a circle.

10. Find the range of values of for which the variable line y 2x lies between the circles

2 2x y 2x 2y 1 0 and 2 2x y 16x 2x 61 0 without intercepting a chord on either

circle.

11. Find the equation of the circle touching the pair of lines 2 27x 18xy 7y 0 and the circle

2 2x y 8x 8y 0, and contained in the given circle.

12. If a vertex of an equilateral triangle is (2 2, 1) and its centroid is (0, 0) then find the

equation of its incircle.

13. P(2, 2) is a point on a circle 2 2x y 2x 2y 0. If P travels on the circle and reaches Q

such that arc 1

PQ6

circumference of the circle then find the coordinates of Q.

14. A regular hexagon is inscribed in the circle 2 2x y 2(x y 1) 0 with one vertex at (3, 1).

Find the coordinates of the two vertices of the hexagon consecutive to the vertex (3, 1).

15. Let 2C be a circle lying inside the circle 1C . The circle C is inside 1C but outside 2C such that

C touches 1C internally but 2C externally. Identify the locus of the centre of C.

www.atharvlabs.com

3

Rapid Revision Sheet Circles

By Pankaj Baluja

Answers

1. (a)

6 8 6 8, ,

5 5 5 5 (b) 2 2x y 4x 2y 4 0 (c) 3570 (d) 2 2x y 4 2x 4 0

2. (a) A (b) A (c) D (d) B (e) A 3. 2 2

x y , x y3 3

4.

3 2 6 3 2 6,

5 5 5.

2 20,

4

6. 2 2 2 2x y 5x 2y 1 0, 2(x y ) 15x y 7 0

8. 2 2x y 4 10. ( 15 2 5, 1 5)

11. 2 2x y 12x 12y 64 0 12. 2 24(x y ) 9

13.

3 3 3 3,

2 2 2 2 14. (2,1 3)

15. Circle if concentric; ellipse if not concentric

www.atharvlabs.com

1

Rapid Revision Sheet Parabola

By Pankaj Baluja

1. Fill in the blanks of the following incomplete statements so that the statements become true.

(a) The focus of a parabola is (0, 2) and the ends of its double ordinate through the focus

are (–1, 2) and (1, 2). Then the equation of the parabola is ____

(b) The general equation to a system of parallel chords of the parabola 2y 4x is

y 2x k. The equation of the corresponding diameter is ______

2. IN the following, each question has four options of which one or more options are correct.

Indicate the correct option(s):

(a) In the normals to the parabola 2y 8ax at the points 1 1 2 2(x , y ), (x , y ) and 3 3(x , y ) are

concurrent then

(A) 1 2 3x x x 0 (B) 1 2 3y y y 0

(C) 1 1 2 2 3 3x y x y x y 0 (D) None of these

(b) S is the focus of a parabola. PQ is any focal chord of the parabola. Then the semilatus

rectum of the parabola is

(A) AM of PS and QS (B) GM of PS and QS

(C) HM of PS and QS (D) None of these (c) P is the point ' t ' on the parabola 2y 4ax and PQ is a focal chord. PT is the tangent at

P and QN is the normal at Q. If the angle between PT and QN be and the distance

between PT and QN be d then

(A) 0 90 (B) 0

(C) d 0 (D)

2 3/2

2

a(1 t )d

t

3. Find the equation of the normal to the parabola 2y 4x at the point (4, 4). Also find the

point on this normal from which the other two normals drawn to the parabola will be at right

angles.

4. Find the equation of the parabola whose axis is along x-axis and which touches the pair of

lines 2 2x y 2x 1 0, focus being at (4, 0).

5. What is the radius of the circle whose centre is (–4, 0) and which cuts the parabola 2y 8x

at A and B such that the common chord AB subtends a right angle at the vertex of the

parabola ?

6. Find the number of points with integral coordinates (2a,a 1) that fall in the interior of the

larger segment of the circle 2 2x y 25 cut off by the parabola 2x 4y 0. Also find their

coordinates.

7. Prove that the point ( , ) which satisfies the equation 1 1

x y 12 3

lies on the part of a

parabola touching the positive x and y axes at distance 4 and 9 respectively from the origin

which falls in the rectangle bounded by the lines x 0, x 4, y 0, y 9.

www.atharvlabs.com

2

Rapid Revision Sheet Parabola

By Pankaj Baluja

8. The equation of a family of straight lines 2y 2at t(x at ) where t is a parameter. Prove

that the locus of the points of intersection of pairs of perpendicular lines of the family is a

parabola. Find the vertex, focus and the latus rectum of the parabola.

9. Prove that the foot of any perpendicular from the point (0, c),c 0, to any normal to the

parabola 2x 4ay lies on the curve whose equation is 4 2 2x (y c){x (2a y) a(y c) }.

10. S is the focus and V is the vertex of the parabola 2x 4ax. P is any point on the parabola.

Prove that the circle drawn on SP as diameter touches the tangent at V to the parabola.

11. Find the equation of the common tangent to the parabolas 2y 32x and 2x 108y.

12. P, Q, R are three points on the parabola 2y 4ax such that the ordinates of the points are in

AP. If ar 2( PQR) 2a , prove that the centroid of the PQR always lies on a parabola having

the same latus rectum.

13. Let P be a fixed point. Any line through P cuts a parabola at Q and R. If S is a point on the

line PQR such that PS is the HM between PQ and PR then prove that the locus of S is a

straight lines whose slope is independent of the abscissa of the point P.

14. Consider the two parabolas 21C : x 1 y and 2

2C : y 1 x. Let P and Q be any two points

on the parabolas 1C and 2C respectively. Let P' and Q' be the reflections of P and Q

respectively with respect to the line y x. Prove that P' lies on 2C ,Q ' lies on 1C and

PQ min{PP ', QQ'}. Also determine the points P, Q such that PQ is the minimum.

15. P, Q and R be three feet of normals to the parabola 2y x 4y 5 drawn from any point.

Prove that the circumcircle of the PQR always passes through a fixed point. Also prove that

the fixed point is the vertex of the parabola.

Answers

1. (a) 2x 2y 3, 2y 5 (b) y 1 2. (a) B (b) C (c) B, D

3. y 2x 12,(7, 2) 4. 2y 6x 15

5. 4 13 6. two; (2, 0), (4, 1)

8.

13a(3a,0), ,0

4 and a respectively 11. 2x 3y 36 0

14.

5 1 1 5Q , , P ,

4 2 2 4

www.atharvlabs.com

1

Rapid Revision Sheet Ellipse and Hyperbola

By Pankaj Baluja

1. Fill in the following incomplete sentences so that the sentences becomes true.

(a) If P and Q are the ends of a pair of conjugate diameters and C is the centre of the

ellipse 2 24x 9y 36 then the area of the CPQ is ____

(b) The angle of intersection of the ellipse 2 2x 4y 4 and the parabola 2x 1 y is ___

(c) If any point on a hyperbola is (3 tan , 2 sec ) then the eccentricity of the hyperbola is

____

2. IN the following, each question has four options of which one or more options are correct.

Indicate the correct option(s).

(a) The normal to the rectangular hyperbola 2xy c at the point ' 1t ' meets the curve again

at the point ' 2t '. Then the value of 31 2t t is

(A) 1 (B) c

(C) –c (D) –1

(b) If 2

cos3

then the range of values of for which the point ' ' on the ellipse

2 2x 4y 4 falls inside the circle 2 2x y 4x 3 0 is

(A) ( , ) (B) (0, )

(C) ( , ) (D) None of these (c) The set of real values of for which the equation

2 2(1 )x 2 xy ( 2)y 4x 3 0 represents a hyperbola is

(A) ( 2, ) (B) ( , 2)

(C) 2 2

2, ,7 7

(D) None of these

3. Tangents at right angles are drawn to the ellipse 2 2x 4y 16. find the equation of the locus

of the middle point of the chord of contact of the tangents.

4. From the point P(1, 3) tangents PA and PB are drawn to the ellipse 2 24x 9y 36. Find the

area of the quadrilateral PAOB where O is the centre of the ellipse.

5. If P and Q are two ends of conjugate diameters and R is the foot of the perpendicular to the

chord PQ from the centre of the ellipse 2 29x 25y 225 then find the equation of the locus

of R.

6. For what real values of a, the point (a,a 2) falls in the interior of the ellipse 2 24x 9y 36

as well as the parabola 2y x ?

7. The sides of a triangle touch the circle 2 2 2x y a and two of the vertices are on the lines

y b. Prove that the locus of the third vertex is 2 2 2

2 2 22 2 2

4a b xx y a .

(a b )

8. Prove that the locus of the middle points of normal chords of the rectangular hyperbola 2 2 2x y a is 2 2 3 2 2 2(y x ) 4a x y .

www.atharvlabs.com

2

Rapid Revision Sheet Ellipse and Hyperbola

By Pankaj Baluja

9. Prove that 2 2

2 2 2

(a b )my mx

a b m

is a normal to the ellipse

2 2

2 2

x y1

a b for all m R. Hence or

otherwise prove that the sum of the angles that the four normals drawn from the point ( , )

to the ellipse 2 2x 2y 4 make with the major axis is equal to the sum of the angles that the

two tangents from the same point make with the major axis.

10. A rectangular hyperbola whose centre is C is cut by a circle of radius r in four points P, Q, R,

S. Prove that 2 2 2 2 2CP CQ CR CS 4r .

11. Let P, Q be two points on the ellipse 2 2

2 2

x y1

a b whose eccentric angles differ by a right

angle. The tangents at P and Q meet at R. Prove that the chord PQ bisects the line segment

CR where C is the centre of the ellipse.

12. Any normal to the hyperbola 2 2

2 2

x y1

a b meets its axes in M and N. The rectangle OMPN is

completed where O is the centre of the hyperbola. Prove that the locus of P is2 2 2 2 2 2 2a x b y (a b ) .

13. P, Q, R are points on a rectangular hyperbola, and PQPR. Prove that the tangent at P is

perpendicular to QR.

14. Let P be a point such that the sum of the slopes of normals drawn from the point P to the

rectangular hyperbola 2xy c is equal to the sum of ordinates of the feet of normals. Prove

that the locus of P is a parabola. Find its focus and latus rectum.

15. Let A, B and C be vertices of an equilateral triangle inscribed in the circle of radius a, the

origin being at the centre. Perpendiculars are drawn from A, B and C to the major axis of the

ellipse 2 2

2 2

x y1,(a b)

a b and they meet the ellipse at P, Q and R respectively. If P, Q and R

lie on the same side of the major axis as A, B and C respectively then prove that the normals

to the ellipse drawn at the point P, Q and R are concurrent.

[Note If A (acos ,a sin ) then P (acos ,b sin ). ]

Answers

1. (a) 23 unit (b) 0° (c) 132

2. (a) D (b) A (c) C

3. 22 2

2 2 x yx y 20

16 4

4. 21793

unit340

5. 2 2 2 2 225x 9y 2(x y ) 6. 36

1,13

14. focus 2c

0, ,4

latus rectum 2c