circl

Pop

ade’

s R

elia

nce

Mat

hs A

cade

my

Par

bhan

i

CIRCLE

Definition :- The locus of a point in a plane at a

constant distance from the fixed point is called as the circle.

The fixed point is called as the centre and the constant

distance is called as the radius of the circle.

Equation of the cirle in various forms :-

1) Standard form :-

The circle whose centre is at origin is called as the stan-

dard circle. The equation of circle with centre at origin

and radius ‘a’ is x2 + y2 = a2.

2) Centre - Radius form :-

The equation of circle with centre at (h, k) and radius r is

(x - h)2 + (y - k)2 = r2.

3) Diameter form :-

The equation of circle whose end points of the diameter are

(x1, y

1) and (x

2, y

2) is

(x - x1)(x - x

2) + (y - y

1)(y - y

2) = 0

4) General form :-

The equation x2 + y2 + 2gx + 2fy + c = 0 represents the

general equation of circle with centre at (-g, -f) and radius

√ √ √ √ √ g2 + f2 - c .

5) Parametric Equations of the circle :-

(i) The parametric equations of the circle x2 + y2 = a2 are

x = acosθθθθθ and y = asinθθθθθ. where θ is called as the param-

eter of any point on the circle.Thus any point on the circle

is P(θ) ≡ (acosθ, asinθ).

(ii) The parametric equations of the circle :

(x - h)2 + (y - k)2 = a2 are

x = h + acosθθθθθ and y = k + asinθθθθθ. where θ is called as

the parameter of any point on the circle. Thus any point

on the circle is P(θ) ≡ (h + acosθ, k + asinθ).

Condition for the general second degree equation to

represent the circle :-

The general second degree equation ax2 + 2hxy + by2 + 2gx

+ 2fy + c = 0 represents the circle, if (i) a = b and (ii) h = 0.

Two Circles Touching Each Other :-

If C1 and C

2 are the centres and r

1, r

2 are the radii of two

circles, then

(i) The circles touch each other externally, if C1C

2 = r

1 + r

2,

and the point of contact divides the segment joining C1 and

C2 internally in the ratio r

1 : r

2.

ii) The circles touch each other internally, if C1C

2 = | r

1 - r

2 |

and the point of contact divides the segment joining C1 and

C2 externally in the ratio r

1 : r

2.

iii) The circles intersect at two points, if

| r1 - r

2 | < C

1C

2 < r

1 + r

2 .

iv) The circles neither intersect nor touch each other, if

C1C

2> r

1+ r

2 or C

1C

2 < | r

1 - r

2|.

Position Of a Point w.r.t. a Circle :-

The point P(x1, y

1) lies outside, on or inside a circle

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

S1 ≡ x

12 + y

12 + 2gx

1 + 2fy

1 + c > = or < 0

Tangents to a Circle :-

Tangent at a given point of the circle :-

1) The equation of tangent to the circle x2 + y2 = a2 at the

point (x1, y

1) of it is x x

1 + y y

1 = a2.

2) The equation of tangent to the circle x2 + y2 + 2gx + 2fy

+ c = 0 at the point (x1, y

1) of it is

x x1 + y y

1 + g (x + x

1) + f (y + y

1) + c = 0.

and the point of contact of this tangent is

( , + )

Orthogonal Circles :-

Two circles are said to be orthogonal, if the tangents at their

point of intersection are at right angles, i.e. r1

2 + r2

2 = d2 , where

r1, r

2 are the radii and d is the distance between two centres of

the circles.

The two circles x2 + y2 + 2g1x + 2f

1y + c

1 = 0 and x2 + y2 +

2g2x + 2f

2y + c

2 = 0 cut each other orthogonally, if

2g1g

2 + 2f

1f

2 = c

1 + c

2.

Director Circle :-

The locus of a point, the tangents from which to the circle are

perpendicular to each other is called as the director circle. The

equation of the director circle of the circle x2 + y2 = a2 is

x2 + y2 = 2a2.

Concentric Circle :-

The circles having the same centre are called as the concentric

circles.

The equation of the circle concentric with the circle x2 + y2 +

2gx + 2fy + c = 0 is x2 + y2 + 2gx + 2fy + k = 0, where k is any

real number.

Length of the tangent segment :

1. The length of the tangent segment drawn to the circle

x2 + y2 = a2 from the point (x1, y

1) is S = √ x

1

2 + y1

2 - a2

2. The length of the tangent segment drawn to the circle

x2 + y2 + 2gx + 2fy + c = 0 from the point (x1, y

1) is

S = √ x1

2 + y1

2 + 2gx1 + 2fy

1 + c

Points to Remember :

1. The lengths of the intercepts on the coordinate axes made by

the general circle are 2 √g2 - c and 2 √f2 -c

2. If OA and OB are the tangents from the origin to the circle

x2 + y2 + 2gx + 2fy + c = 0 and C is the centre of the circle,

then the area of the quadrilateral OACB is √c(g2 + f2 - c)

3. The angle between the tangents from (x1, y

1) to the circle

x2 + y2 = a2 is

2 tan-1( )

4. If two tangents drawn from the origin to the circle x2 + y2 +

2gx + 2fy + c = 0 are perpendicular to each other, then

g2 + f2 = 2c

5. The length of the tangent drawn from any point on the circle

x2 + y2 + 2gx + 2fy + c1 = 0 to the circle x2 + y2 + 2gx + 2fy +

c = 0 is √c - c1.

6. If the radius of the circle x2 + y2 + 2gx + 2fy + c = 0 be r

and it touches both the axes, then g = f = √c = r

7. If the circles x2 + y2 + 2gx + c2 = 0 and x2 + y2+2fy + c2 = 0

touch each other, then

+ =

S.M.Popade

ma

√1 + m2+ a

√1 + m2

Tangent in Parametric form : -

The equation of tangent to the circle x2 + y2 = a2 at the point

P(θ) is xcosθθθθθ + y sinθθθθθ = a

Condition for Tangency and Equation of Tangent in terms

of Slope :-

The line y = mx + c is a tangent to the circle x2 + y2 = a2, if c2

= a2 (m2 + 1) and the equation of tangent in terms of slope is

y = mx + a √ 1 + m2.

a

√x1

2 + y1

2 - a2

1

g2

1

f2

1

c2

Pop

ade’

s R

elia

nce

Mat

hs A

cade

my

Par

bhan

i

1. The area of circle centred at (1, 2) and passing through (4, 6)

a) 5π . b) 10π c) 25π d) None of these.

2. If ( -3, 2) lies on the circle x2 + y2 + 2gx + 2fy + c = 0

which is concentric with the circle x2 + y2 + 6x + 8y - 5 = 0.

then c is

a) 11 b) -11

c) 24 d) None of these.

3. Equation of a circle through (- 1. - 2) and concentric with the

circle x2 + y2 - 3x + 4y - c = 0 is

a) x2 + y2- 3x + 4y - 1 = 0 b) x2 + y2 - 3x + 4y = 0

c) x2 + y2 - 3x + 4y + 2 = 0 d) None of these.

4 The equation of a circle passing through the point (4, 5)

having the centre at (2. 2) is

a) x2 + y2 + 4x + 4y - 5 = 0 b) x2 + y2 - 4x - 4y - 5 = 0

c) x2 + y2 - 4x = 13 d) x2 + y2 - 4x - 4y + 5 = 0.

5. If the line y = x + 3 meets the cilcle x2 + y2 = a2 at A and B,

then equation of the circle on AB as diameter is

a) x2 + y2 + 3x - 3y - a2 + 9 = 0

b) x2 + y2 - 3x + 3y - a2 + 9 = 0

c) x2 + y2 + 3x + 3y - a2 + 9 = 0

d) None of these.

6. If the coordinates at one end of a diameter of the circle

x2 + y2 - 8x - 4y + c = 0 are (- 3, 2), then the coordinates at the

other end are

a) (5, 3) b) (6. 2)

c) (l. - 8) d) (11,2).

7. Equation of the circle through origin which cuts intercepts

of length a and b on axes is

a) x2 + y2 + ax + by = 0 b) x2 + y2 - ax - by = 0

c) x2 + y2 + bx + ay = 0 d) None of these.

8. Equation of the diameter of the circle x2 + y2 - 2x + 4y = 0

which passes through the origin is

a) x + 2y = 0 b) x - 2y = 0

c) 2x + y = 0 d) 2x - y = 0.

9. The line x + 3y = 0 is the diameter of the circle

a) x2 + y2 + 6x + 2y = 0. b) x2 + y2- 6x + 2y = 0

c) x2 + y2 - 6x - 2y = 0 d) x2 + y2 + 8x - 2y = 0.

10. The equation ax2 + by2 + 2hxy + 2gx + 2fy + c = 0

represents a circle, the condition will be

a) a = b and.c = 0 b) f = g and h = 0

c) a = b and h = 0 d) f = g and c = 0.

11. Length of the chord on the line 4x - 3y - 10 = 0 cut off by the

circle x2 + y2 - 2x + 4y - 20 = 0 is

a) 10 b) 6


12. From origin, chords are drawn to the circle x2 + y2 -2y = 0

The locus of the middle points of these chords is

a) x2 + y2 - y = 0 b) x2 + y2 - x = 0

c) x2 + y2 - 2x = 0 d) x2 + y2 - x - y = 0

13. A square is inscribed in the circle x2 + y2 - 2x + 4y + 3 = 0

Its sides are parallel to the coordinate axes. Then one

vertex of the square is

a) (1 +√2, - 2 ) b) (1 - √2, - 2 )

c) (1, -2 +√2 ) c) (2, -3)

14.The equation of the circumcircle of the triangle formed by

the lines y + √3x = 6, y - √3x = 6 and y = 0 is

a) x2 + y2 - 4y = 0 . b) x2 + y2 + 4x = 0

c) x2 + y2 - 4y = 12 d) x2 + y2 + 4x = 12.

15. Locus of a point such that the ratio of its distances from

two fixed points is constant is

a) a circle b) a straight line

c) an ellipse d) None of these.

16. The circles x2 + y2 - 10x + 16 = 0 and x2 + y2 = r2 intersect

each other in two distinct points if

a) r < 2 b) r > 8

c) 2 < r < 8 d) 2 < r < 8.

17. The length of the intercept cut by the circle

x2 + y2 - 7x + 5y + 12 = 0 on x-axis is equal to

a) 3 b) 1

c) 4 d) 2.

18. A circle is given by x2 + y2 - 6x + 8y - 11 = 0 and there are

two points (0, 0) and (1, 8). These points lie

a) one outside and one inside the circle

b) both inside the circle

d) both outside the circle

d) one on and the other inside the circle

19. If a circle passes through the points of intersection of the

coordinate axes with the lines lx - y + 1 = 0 and x -2y +

3 = 0.then the velue of l is

a) 2 b) 1/3

c) 6 d) 3

20. The number of integral values of k for which x2 + y2 + kx +

(1 - k) y + 5 = 0 represents a circle whose radius cannot

exceed 5, is :

a) 14 b) 16

c) 18 d) 20.

21. Given the circles x2+ y2 - 4x- 5 = 0 and x2+ y2 + 6x - 2y +

6 = 0. Let P be a point (α,β) such that the tangents from P

on both the circles are equal in length. Then

a) 2α + 10β + 11 = 0 b)2α - 10β + 11 = 0

c) l0α-2β+ 11 = 0 d) l0α+2β+ 11 = 0.

22.Range of values of m for which the straight line y = mx + 2

cuts the circle x2 + y2 = 1 in distinct or coincident points is :

a) [-√3, √3 ] b) [√3, ∞)

c) (-∞, -√3 ] ∪ [√3, ∞) d) None of these

23. The equation of the tangents drawn from the origin to the

circle x2 + y2 - 2rx - 2hy + h2 = 0 are

a) x = 0, b) y = 0

c) (h2 - r2)x - 2hry = 0 d) (h2 - r2)x + 2hry = 0

24. The two circles x2 + y2 - 10x + 4y - 20 = 0 and x2 + y2 +

14x - 6y + 22 = 0 are related by

a) touch externally b) intersect in real points

c) donot intersect d) one is contained in other

25. The tangent to the circle x2 + y2 = 5 at the point (1, -2) also

touches the circle x2 + y2 - 8x + 6y + 20 = 0 at

a) (-2, 1) b) (-3, 0)

c) (-1, -1) d) (3, -1)

26. Image of the circle x2 + y2 + 14x + 6y + 42 = 0 in the line

2x + 3y = 16 is

a) x2 + y 2 + 2x - 12y + 21 = 0

b) x2 + y 2 - 2x - 12y + 21 = 0

c) x2 + y 2 - 2x + 12y + 21 = 0

d) x2 + y 2 + 2x - 12y - 21 = 0

27. The equation of the circle which touches the y-axis at a

distance 4 from the origin and intercepts a length 6 on x-

axis is

a) x2 + y2 + 10x - 8y + 10 = 0 b) x2 + y2 + 10x - 8y - 16 = 0

c) x2 + y2 + 10x + 8y + 16 = 0 d) None of these

CIRCLE

Multiple Choice Questions

Believe on your strength

Popade’s Reliance Maths Academy, Parbhani

9822772673

Pop

ade’

s R

elia

nce

Mat

hs A

cade

my

Par

bhan

i

28. The length of the tangent from (5, 1) to the circle x2 + y2 +

6x - 4y - 3 = 0, is

a) 81 b) 29

c) 7 d) 21

29. The equation of the circle passing through (4, 5) and having

the centre at (2, 2) is

a) x2 + y2 + 4x + 4y - 5 = 0 b) x2 + y2 - 4x - 4y - 5 = 0

c) x2 + y2 - 4x = 13 d) x2 + y2 - 4x - 4y + 5 = 0

30. The equation of the tangents drawn from the origin to the

circle x2 + y2 - 2rx - 2hy + h2 = 0 are

a) x = 0, b) y = 0

c) (h2 - r2)x - 2hry = 0 d) (h2 - r2)x + 2hry = 0

31. The two circles x2 + y2 - 10x + 4y - 20 = 0 and x2 + y2 +

14x - 6y + 22 = 0 are related by

a) touch externally b) intersect in real points

c) donot intersect d) one is contained in other

32. The tangent to the circle x2 + y2 = 5 at the point (1, -2) also

touches the circle x2 + y2 - 8x + 6y + 20 = 0 at

a) (-2, 1) b) (-3, 0)

c) (-1, -1) d) (3, -1)

33. The lengths of the intercept made by the circle x2 + y2 - 5x -

13y - 14 = 0 on X and Y axes are

a) 9, 13 b) 5, 13

c) 9, 15 d) None of these

34. The equation of the circle which touches the y-axis at a

distance 4 from the origin and intercepts a length 6 on x-

axis is

a) x2 + y2 + 10x - 8y + 10 = 0 b) x2 + y2 + 10x - 8y - 16 = 0

d) x2 + y2 + 10x + 8y + 16 = 0 d) None of these

35. Centre of a circle is (2, 3). If the line x + y = 1 touches it,

its equation is

a) x2 + y2 - 4x - 6y + 4 = 0 b) x2 + y2 - 4x - 6y + 5 = 0

d) x2 + y2 - 4x - 6y - 5 = 0 d) None of these

36. The tangent to the circle x2 + y2 = 169 at (5, 12) & (12, -5)

are

a) parallel b) Perpendicular

c) Coincide d) None of these

37. The locus of the point of intersection of the perpendicular

tangents of x2 + y2 = 4 is

a) x2 + y2 = 8 b) x2 + y2 = 12

c) x2 + y2 = 16 d) x2 + y2 = 4√3

38. The equation of the circle passing through (2, 1) and

touching the coordinate axes is

a) x2 + y2 - 2x - 2y + 1 = 0 b) x2 + y2 + 2x + 2y + 1 = 0

d) x2 + y2 - 2x - 2y - 1 = 0 d) x2 + y2 + 2x + 2y - 1 = 0

39. Circle which passes through (1, -2) and (4, -3) and whose

centre lies on 3x + 4y = 7 is

a) x2 + y2 + 3x + 12y + 2 = 0 b) x2 + y2 + 12x + 3y + 2 = 0

c) x2 + y2 - 3x - 12y + 2 = 0 d) x2 + y2 - 12x - 3y + 2 = 0

40. The length of the tangent from the point (1, 2) to the circle

2x2 + 2y2 + 6x - 8y + 3 = 0 is

a) 3/√2 b) √3 / 2

c) √(3/2) d) None of these

41.The line y = mx + c touches the circle x2 + y2 = a2 at

a) ( , ) b) ( , )

c) ( , ) d) ( , )

42. The locus of the point of intersection of two perpendicular

tangents to the circle is

(a) Auxilliary Circle (b) Director Circle

(c) Straight line (d) None of these

43. The radius of the circle through (2, 3) , (-1, 6) and having

centre on 2x + 5y + 1 = 0 is

(a) √13 (b) √23

(c) √29 (d) √31

-a2m

c

-a2

c

-a2m

c

a2

c

a2m

c

a2

c

-a2m

c

-a2

c

45. The slope of the tangent at the point (h, h) to the circle

x2 + y2 = a2 is

a) 0 (b) 1

c) -1 d) Depends on h.

46. The equations of the tangents to the circle x2 + y2 = 4

which are inclined at 600 with x-axis

a) √3 x - y + 3 = 0 b) √3 y - x + 3 = 0

c) √3 x - y + 4 = 0 d) None of these

47. The line 3x + 4y = k touches the circle x2 + y2 = 10x, then

k is

a) -40 b) 10

c) 40 b) -10

48. The radius of the circle 4x2 + 4y2 - 10x + 5y + 5 = 0 is

a) 3√5/8 b) 3√5/7

c) 3√5 d) 5

49. The point of contact of the two circles

x2 + y2 + 2x + 8y - 23 = 0 and x2 + y2 - 4x - 10y + 19 = 0

divides the segment C1C

2 in the ratio

a) 3 : 1 b) 1 : 2

c) 2 : 1 d) 1 : 3

50. The equation of the circle whose centre is (-4, 2) and its

tangent is x - y = 3 is

a) x2 + y2 + 8x - 4y - 41 = 0

b) 2x2 + 2y2 + 16x - 4y + 20 = 0

c) 2x2 + 2y2 + 16x - 8y - 41 = 0

d) x2 + y2 + 16x - 4y - 24 = 0

51. The equations of the tangents drawn from the point (0, 1)

to the circle x2 + y2 - 2x + 4y = 0

a) 2x - y - 1 = 0, x - 2y + 2 = 0

b) 2x - y + 1 = 0, x + 2y - 2 = 0

c) x - y + 1 = 0, x + y + 2 = 0

d) None of these.

52. The equation of the circle with centre at (a, a) and radius‘a’

a) x2 + y2 + 2ax + 2ay + a2 = 0

b) x2 + y2 = a2

c) x2 + y2 - 2ax - 2ay + a2 = 0

d) None of these.

53. The centre and radius of the circle x2 + y2 = 2 is

a) (0, 0); 2 b) (2, 2); 2

c) (0, 0); 4 d) (0, 0); √2

54. The centre and radius of circle 2x2 + 2y2 - 6x + 4y - 3 = 0

is

a) (3/2, -1); 19/2 b) (3/2, -1) √19/2

c) (-3/2, 1); 19/2 d) (-3/2, 1) √19/2

55. The equations of circles with radius ‘r’ and touching the

x- axis at (h, 0)

a) x2 + y2 - 2hx - 2ry + h2 = 0, x2 + y2 - 2hx + 2ry + h2 = 0

b) x2 + y2 + 2hx + 2ry + h2 = 0, x2 + y2 - 2hx - 2ry + h2 = 0

c) x2 + y2 + 2hx + 2ry + h2 - r2 = 0, x2 + y2 - 2hx - 2ry = 0

d) None of these

56. The centre and radius of the circle

(x - 5)(x - 3) + (y - 2)(y - 4) = 0 is

a) (3, 4); √2 b) (4, 3) ; 2

c) (4, 3); √2 d) (5, 3) ; 4

57. The equation of the circle with centre at (3, -2) and touch-

ing the x-axis is

a) x2 + y2 - 6x + 4y + 13 = 0 b) x2 + y2 - 3x + 2y + 9 = 0

c) x2 + y2 + 6x - 4y + 9 = 0 d) x2 + y2 - 6x + 4y + 9 = 0

58. The equation of the circle through the points (4, 1), (-3, -6)

and (-2, 1) is

a) x2 + y2 - 2x + 6y - 15 = 0 b) x2 + y2 - 4x - 5y + 9 = 0

c) x2 + y2 + 4x + 4y + 7 = 0 d) x2 + y2 + 2x - 6y - 15 = 0

59. The values of ‘a’ and ‘h’ for which the equation,

ax2 + hxy + 2y2 + 16x - 8y + 5 = 0 represents a circle

a) a = 0, h = 0 b) a = 2, h = 2

c) a = 5, h = √2 d) None of these

60. If (-3, 2) lies on the circle x2 + y2 + 2gx + 2fy + c = 0 which

is concentrc with x2 + y2 + 6x + 8y - 5 = 0, then c is

a) 11 b) -11

c) 24 d) None of these

44. The equation x2 + y2 + 2gx + 2fy + c = 0 represents a

circle , if

(a) g2 + f 2 - c < 0 (b) g2 + f 2 - c > 0

(c) g2 + f 2 - c = 0 (b) g = f

RMA PBN

Pop

ade’

s R

elia

nce

Mat

hs A

cade

my

Par

bhan

i

61. The length of the intercept cut off by the circle x2 + y2 _ 6x

- 4y - 12 = 0 on the line 4x - 3y + 5 = 0 is

a) (2√126)/5 b) (4√126)/5

c) (√126)/5 d) None of these.

62. The locus of the mid-points of the chords of the circle

x2 + y2 = 4 which subtends a right angle at the origin is

a) x + y = 2 b) x2 + y2 = 1

c) x2 + y2 = 2 d) x + y = 1

63. The equation of the circle having its centre on the line

x + 2y - 3 = 0, and passing through the points of

intersection of the circles

x2 + y2 - 2x - 4y + 1 = 0

and x2 + y2 - 4x - 2y + 4 = 0 is

a) x2 + y2 - 6x + 7 = 0 b) x2 + y2 - 3x + 4 =0

c) x2 + y2 - 2x - 2y + 1 =0 d) x2 + y2 +2x - 4y + 4 = 0.

64. Given two circles x2 + y2 - 4x - 5 = 0, x2 + y2 6y + 8 = 0

Let P (h, k) be a point such that the tangents from P to both

circles are equal in length, then

a) 4h - 6k + 13 = 0 b) 4h - 6k - 13 = 0

c) 4x + 6y - 3 = 0 d) None of these.

65. The circles x2 + y2- 4x + 6y + 8 = 0 and

x2 + y2 - l0x - 6y + 14 = 0,

a) touch externally b) touch internally

c) intersect d) Do not touch.

66. Two circles x2 + y2 = 6 and x2 + y2 - 6x + 8 = 0

are given. Then the equation of the circle through their points

of intersection and the point (1, 1) is

a) x2 + y2 - 6x + 4 = 0 b) x2 + y2 - 3x + 1 = 0

c) x2 + y2 - 4y + 2 = 0 d) None of these.

67. If the two circles x2 + y2 + 2gx + 2fy = 0

and x2 + y2 + 2g| x + 2f | y = 0 touch each other, then

a) fg = f | g| b)f | g = f g|

c) f f | = g g| d) None of these.

68. The condition for the two circles x2 + y2 + 2k1x + k2 = 0

and x2 + y2 + 2k2x + k2 = 0 to touch each other externally is

a) k1

2 + k2

2 = k2 b) k1

2 - k2

2 = k2

c) k2 ( k1

2 - k2

2 ) = k1

2.k2

2 d) k2 ( k1

2 + k2

2 ) = k1

2.k2

2

69. The circles (x - 1)2 + (y - 2)2 = 16 and (x + 4)2 + (y + 3)2 = 1

a) touch b) are concentric

c) intersect d) None of these.

70. The lines 3x - 4y + 4 = 0 and 3x - 4y - 5 = 0 are tangents to

the same circle. The radius of this circle is

a) 9/5 b) 9/10

c) 1/5 d) 1/10.

71. If the lines 2x - 4y = 9 and 6x - 12y + 7 = 0 touch a circle,

then the radius of the circle is

a) √3/5 b) 17/6√5

c) 2√5/3 d) 17/3√5

72.The equation of the line passing through the points of inter-

section of the circles x2 + y2 - 2x - 4y - 4 = 0 and x2 + y2 - 8x

- 12y + 51 = 0 is

a) 6x + 8y - 55 = 0 b) 6x + 8y + 55 = 0

c) No such line d) None of these.

73. If 3x + y = 0 is a tangent to the circle which has its centre at

the point (2. - 1), then the equation of the other tangent to the

circle from the origin is

a) x - 3y = 0 b) x + 3y = 0

c) 3x- Y = 0 d) x + 2y = 0.

74.Equation of circle drawn on the chord 2x + 3y = 13 of the

circle x2 + y2 = 13 as diameter is

a) x2 + y2 - 4x - 6y + 13 = 0 b) x2 + y2 + 4x + 6y + 13 = 0

c) x2 + y2 - 4x + 6y + 13 = 0 d) None of these.

75.Values of p and q for which the equation (x + y - 41)2 - (x +

7y - 7) (px + qy + 1) = 0 represents a circle, are given by

a) p = 7/25, q = 1/25 b) p = 1/25, q = 7/25

c) p = - 7/25, q = -1/25 d) None of these.

76. The number of common tangents to two circles

x2 + y2 = 4 and x2 + y2 - 8x + 12 = 0 is

a) 1 b) 2

c) 3 d) 4.

77. If OA and OB are the tangents from origin to the circle x2 +

y2 + 2gx + 2fy + c = 0 (c < 0) and C is the centre of the circle.

the area of the quadrilateral OACB is

a) 1/2 [√(c (g2 +f 2 - c)] b) [√(c (g2 +f 2 - c)]

c) c√ g2 +f 2 - c d) None of these.

78. The abscissae of two points A and B are the roots of the

equation x2 + 2ax - b2 = 0 and their ordinates are the roots

of the equation x2 + 2px - a2 = 0. The radius of the circle

with AB as diameter is

a) √(a2 + p2) b) √(b2 + q2)

c) √(a2 + b2 + p2 + q2) d) None of these.

79. The area of the triangle formed by joining the origin to the

points of intersection of the line x.√5 + 2y = 3.√5 and circle

x2 + y2 = 10, is

a) 3 b) 4

c) 5 d) 6.

80. The number of common tangents to the circles x2 + y2 = 4

and x2 + y2 - 6x - 8y = 24 is

a) 0 b) 1

c) 3 d) 4

81. The equation of the circle which has tangent 2x - y - 1 = 0

at (3, 5) on it and with the centre on x + y = 5. is

a) x2 + y2 + 6x - 16y + 28 = 0

b) x2 + y2 - 6x + 16y - 28 = 0

c) x2 + y2 + 6x + 6y - 28 = 0

d) x2 + y2 - 6x - 6y - 28 = 0.

82. The circles x2 + y2 + 4x + 6y - 8 = 0 and

x2 + y2 + 6x - 8y + c = 0, cut orthogonally, if c =

a) -4 b) 4

c) 2 d) -2

83. If the circles of same radius ‘a’ and centres at (2, 3) and

(5, 6) cut orthogonally, then a =

a) 3 b) -3

c) 4 d) 2

84. The equation 2x2 + 2y2 - 6x + 8y + k = 0 represents a point

circle, if k is

a) 2/25 b) 25

c) 25/2 d) 2

85. Equation of the circle having diametres 2x - 3y = 5 and

3x - 4y = 7 and radius 8, is

a) (x - 1)2 + (y - 1)2 = 82 b) (x - 1)2 + (y + 1)2 = 82

c) (x + 1)2 + (y - 1)2 = 82 d) (x + 1)2 + (y + 1)2 = 82

86. The vertices of a right angled triangles are A(2, -2), B(-2, 1)

and C(5, 2). The equation of circumcircle is,

a) (x + 5)(x - 5) + (y - 1)(y - 2) = 0

b) (x + 5)(x + 5) + (y - 1)(y - 2) = 0

c) (x + 5)(x - 5) + (y - 1)(y + 2) = 0

d) None of these.

87. Centre of the circle (x - a)(x - b) + (y - p)(y - q) = 0, is

a) (a/2, b/2) b) ( , )

c) (p/2, q/2) d) ( , )

88. The angle between the tangents drawn from the point

(13, 0) to the circle x2 + y2 = 25 are

a) tan-1(5/12) b) 2tan-1(5/12)

c) tan-1(12/5) d) None of these

89. The equations of the tangents drawn from the point (0, 1) to

the circle x2 + y2 - 2x + 4y = 0 are

a) 2x - y - 1 = 0, x - 2y + 2 = 0

b) 2x - y + 1 = 0, x + 2y - 2 = 0

c) x - y + 1 = 0, x + y + 2 = 0

d) None of these.

90. A circle with radius 12 lies in the first quadrant and

touches both the axes, another circle has its centre at (8, 9)

and radius 7. Which of the following is true

a) circles touch each other internally

b) circles touch each other externally

c) circles intersect each other at two points

d) None of these

a - b

2

p - q

2a + b

2

p + q

2

RMA PBN

Pop

ade’

s R

elia

nce

Mat

hs A

cade

my

Par

bhan

i

95. The radius of the circle inscribed in the triangle formed by

the lines x = 0, y = 0 and 4x + 3y - 24 = 0 is :

a) 12 b) 2

c) 2/3 d) 6.

96. The equation of the chord of the circle x2 + y2 - 4x = 0,

whose mid-point is (1, 0) is :

a) y = 2 b) y = 1

c) x = 2 d) x = 1.

97. A variable circle passes through the fixed point A (p, q) and

touches x-axis. The locus of the other end of the diameter

through A is :

a) (x - p)2 = 4qy b) (x - q)2 = 4py

c) (y - p)2 = 4qx d) (y - q)2 = 4px.

98. The equation of the circle whose radius is 5 and which

touches the circle x2 + y2 - 2x - 4y - 20 = 0 at the point (5, 5)

a) x2 + y2 + 18x + 16y + 120 = 0

b) x2 + y2 - 18x - 16y + 120 = 0

c) x2 + y2 - 18x + 16y + 120 = 0

d) x2 + y2 + 18x - 16y + 120 = 0.

99.The tangent to the circle x2 + y2 = 9, which is parallel to

y-axis and does not lie in the third quadrant, touches the

circle at the point: .

a) (3,0) b) (- 3, 0)

c) (0, 3) d) (0, - 3).

100. A circle passes through the points of intersection of the

line x = 0 and the circle x2 + y2 + 2x = 3. If this circle

passes through (13, 0), then its centre is :

a) (0, 0) b) (0, 1)

c) (1, 0) d) (1, 1).

101.If x = 7 touches the circle x2 + y2 - 4x - 6y - 12 = 0, then the

coordinates of the point of contact are :

a) (7, 3) b) (7, 4)

c) (7, 8) d) (7, 2).

102. The centre of a circle passing through the points (0, 0),

(1, 0) and touching the circle x2 + y2 = 9 is:

a) (3/2, 1/2) b) (1/2, 3/2)

c) (1/2, 1/2) d) (1/2, + √2)

103. If the equation k + = 1 represents a

circle, then the value of k isa) 3/2 b) 1

c) 3/4 d) None of these

104. If the line 2x - y + k = 0 is a diameter of the circle x2 + y2

+ 6x - 6y + 5 = 0, then k is :

a) 6 b) 9


105. The condition that the line (x + g) cosθ + (y + f) sinθ = k

is a tangent to x2 + y2 + 2gx + 2fy + c = 0 :

a) g2 + f 2 = c2 + k b) g2 + f 2 = c + k

c) g2 + f = c + k2 d) None of these.

106. If the straight line 3x + 4y = k touches the circle x2 + y2 -

l0x = 0, then the value of k is:

a) 2 or 20 b) - 2 or 20

c) - 1 or 20 d) - 10 or 40.

107. The triangle PQR is inscribed in the circle x2 + y2 = 25. If

Q and R have coordinates (3, 4) and (- 4, 3) respec-tively,

then ∠QPR is equal to :

a) π/2 b) π/3

c) π/4 d) π/6.

108. If 2x2 + kxy + 2y2 + (k - 4)x + 6y - 5 = 0 represents a circle,

then its radius is

a) 2√2 b) 3√2

c) 2√3 d) None of these

109.The equation of the diameter of the circle x2 + y2 = 2ay that

is perpendicular to the line x + 2y = 4 is :

a) 2x - y + a = 0 b) x + 2y - a = 0

c) 2x - 2y + a = 0 d) None of these

110.The circles x2 + y2-25 = 0 & 3(x2 + y2)- 30x + 40y + 175 = 0

touch each other at

a) (3, 4) b) (1, 2)

c) (4, 5) d) None of these

111. The tangents drawn to the circle x2 + y2 = 10 from the

point (2, -4) include an angle of measure

a) 600 b) 450

c) 300 d) 900

112. The value of k, if the line 2x - 3y + k = 0 is a tangent to

the circle x2 + y2 + 2x - 4y - 8 = 0

a) -21, 5 b) 21, -5

c) + 13 d) + √13

113.Tangents drawn from the point P(3, 4) to the circle x2 + y2

= 16 touch it at Q and R. O is the centre of the circle, then

the area of the quadrilateral OQPR is

a) 6 units b) 24 units

c) 12 units d) 48 units

114. The equation of the circle concentic with x2 + y2 - 6x +

12y + 15 = 0 and having double radius is

a) (x - 3)2 + (y + 6)2 = 120 b) x2 + y2 - 6x + 12y + 45 = 0

c) (x - 3)2 + (y + 6)2 = 45 d) None of these

115. The length of the y-intercept of the circle x2 + y2 + 4x - 7y

+ 12 = 0 is

a) 1 b) 3

c) 4 d) 7

116. If two circles x2 + y2 - 3x + ky - 5 = 0 and 4x2 + 4y2 - 12x

- y - 9 = 0 are concentric, then k =

a) 1/4 b) -1/4

c) 4 d) -4

117. The radius of the circle touching the lines 3x - 4y + 5 = 0

and 6x - 8y - 9 = 0 is

a) 19/10 b) 9/10

c) 1/2 d) None of these.

118. The equation of the diameter of the circle x2 + y2 + 2x - 4y

+ 1 = 0 parallel to x-axis is

a) y = 2 b) y = 4

c) 3x + 2y + 4 = 0 d) y = -2

119. Coordinates of the point from which the lengths of tangents

to the following three circles x2 + y2 = 4, x2 + y2 - 2x - 4y +

1 = 0, x2 + y2 - 4x - 2y + 1 = 0 are equal, are

a) (5/6, 5/6) b) (5/6, 6/5)

c) (5/6, - 5/6) d) (-5/6, 5/6).

120. Equation of the circle which passes through the origin and

cuts orthogonally each of the circles x2 + y2- 8y+ 12= 0 is

x2 + y2 - 4x - 6y - 3 = 0

a) x2 + y2 + 6x + 3y = 0 b) x2 + y2 + 3x- 6y = 0

c) x2 + y2 + 6x - 3y = 0 d) x2 + y2 - 3x + 6y = 0

121. The equation of the common chord of the circles

x2 + y2 - 6x = 0 and x2 + y2 - 4y = 0 is

a) 3x + 2y + 1 = 0 b) 3x - 2y = 0

c) 3x + 2y = 0 d) 3x - 2y - 1 = 0.

122. Equation of common chord of the circles x2 + y2 - 2x - 4y +

1 = 0 and x2 + y2 - l0x + 8y + 25 = 0 is

a) 2x - 3y - 6 = 0 b) 2x + 3y + 6 = 0

c) x - 2y - 3 = 0 d) None of these.

(x + 1)2

3

(y + 2)2

4

“The most important key to achieving great

success is to decide upon your goal and

launch, get started, take action, move.”

123. The number of tangents, which can be drawn from the

point (1, 2) to the circle x2 + y2 = 5, is :

a) 1 b) 2

c) 3 d) 0.

124. Points (2, 0), (0, 1), (4, 5) and (0, a) are concyclic for a =

a) 14/3 or 1 b) 14 or 1/3

c) -14/3 d) None of these

125. The equation of the circle which has two normals (x - 1)(y

- 2) = 0 and a tangent 3x + 4y = 6 is

a) x2 + y2 - 2x - 4y + 4 = 0 b) x2 + y2 + 3x - 4y + 5 = 0

c) x2 + y2 = 5 d) (x - 3)2 + (y - 4)2 = 5

126. The line joining (5, 0) to (10cosθ, 10sinθ) is divided

internally in the ratio 2 : 3 at P. If θ varies, then the locus

of P is

a) a pair of straight line b) a circle

c) a straight line d) None of these

RMA PBN

Pop

ade’

s R

elia

nce

Mat

hs A

cade

my

Par

bhan

i

127. The equation of the circle of radius 5 in the first quadrant

which touches x-axis and the line 4y = 3x is

a) x2 + y2 - 2x - 4y - 20 = 0

b) x2 + y2 + 10x - 30y + 25 = 0

c) x2 + y2 + 30x - 10y - 225 = 0

d) x2 + y2-30x - 10y + 225 = 0

128. The area of the triangle formed by the tangents from the

point (4, 3) to the circle x2 + y2 = 9 and the line joining

their points of contact is

a) 192 b) 192/24


129. The equation of the circle which touches the axes of the

coordinate axes and the line (x/3) + (y/4) = 1 and whose

centre lies in the first quadrant is x2 + y2 - 2cx - 2cy + c2 = 0

then c is

a) 1, 6 b) 2, 3

c) 3, 4 d) 4, 5

130. The locus of the centre of a circle which cuts orthogo-

nally the circle x2 + y2 - 20x + 4 = 0 and which touches

x = 2 is

a) y2 = 16x b) x2 = 16y

c) y2 = 16x + 4 d) x2 = 16y + 4

131.The coordinates of radical centre of the circles x2 + y2 = 9,

x2 + y2 - 2x - 2y = 5 and x2 + y2 + 4x + 6y = 19 is

a) (0, 0) b) (1, 1)

c) (-1, 1) d) (1, -1)

132. The common tangents to the ciercles x2 + y2 + 2x = 0 and

x2 + y2 - 6x = 0 form a triangle which is

a) isosceles b) equilateral

c) right angled d) None of these

133. The circle passing through the intersection of the circle

S = 0 and the line P = 0 is

a) S + kP b) S - k P

c) kS + P d) All the above

134. The area of the circle whose centre is (h, k) and radius is

‘a’ is

a) π b) 2π2

c) πa2 d) None of these

135. The equation of the tangents which make equal intercepts

on the positive coordinate axes is given by

a) x + y + 2√2 = 0 b) x + y = 2√2

c) x + y = 2√3 d) None of these

136. The point at which the line x = 0 touches the circle x2 + y2

- 2x - 6y + 9 = 0 is

a) (0, 2) b) (0, 8)

c) (0, 3) d) (0, 4)

137. For the circle x2 + y2 + 3x + 3y = 0, which of the follow

ing relation is true

a) centre lies on the x -axis b) centre lies on the y-axis

c) centre at origin d) circle pass through origin

138. For the given circles x2 + y2 - 6x - 2y + 1 = 0 and x2 + y2

+ 2x - 8y + 13 = 0, which of the following is true

a) one circle lies inside the other

b) one circle lies completely outside the other

c) two circles intersect in two points

d) They touch each other

139. The points of intersection of the circles x2 + y2 = 25 and

x2 + y2 - 8x + 7 = 0 are

a) (4, 3) and (4, -3) b) (2, -3) and (-3, -2)

c) (-6, 3) and (4, 3) d) (2, 3) and (3, 4)

140. If the radius of the circle x2 + y2 - 18x + 12y + k = 0 is

11, then k =

a) -3 b) 4

c) -4 d) 3

141. The straight line (x - 2) + (y + 3) = 0 cuts the circle

(x - 1)2 + (y - 3)2 = 11 at

a) No point b) one point

c) Two points d) Three points

142.The equation of the circle which touches both the axes

and whose centre is (x1, y

1) is

a) x2 + y2 + 3x1(x + y) + x

1

2 = 0

b) x2 + y2 - 2x1(x + y) + x

1

2 = 0

c) x2 + y2 + 3xx1 + 2yy

1 = 0

d) x2 + y2 = x1

2 + y1

2

143. A circle with radius 12 lies in the first quadrant and

touches both the axes, another circle has its centre at (8, 9)

and radius 7. Which of the following is true

a) circles touch each other internally

b) circles touch each other externally

c) circles intersect each other at two points

d) None of these

144. The locus of the point whose shortest distance from the

circle x2 + y2 - 2x + 6y - 6 = 0 is equal to its distance from

the line x - 3 = 0

a) x2 + 6y - 4x - 9 = 0 b) y2 - 6y + 4x + 9 = 0

c) x2 - 6y - 4x - 9 = 0 d) y2 + 6y - 4x + 9 = 0

145. If the circle x2 + y2 + px + qy + c = 0 touches the x-axis,

then

a) p2 = c b) p2 = 4c

c) q2 = 4c d) None of these

146. The circumference of the circle 9x2 + 9y2 - 12x - 24y -

124 = 0 is

a) 4π b) 8π c) 16π d) 20π147.The locus of the centre of the circle which touches the x-

axis and cuts off an intercept of constant length k from the

y-axis is

a) x2 + y2 = k2/4 b) x2 - y2 = k2/4

c) y2 - x2 = k2/4 d) x2 - y2 = k2

148. A(4, 1) is a point on the circle x2 + y2 - 2x + 6y - 15 = 0,

then the equation of the tangent to the circle, which is

parallel to the tangent at A is

a) 3x + y = 0 b) 3x + 4y - 16 = 0

c) 3x - 10y = 34 d) 3x + 4y + 34 = 0

149. The area of the largest rectangle which can inscribed in

the circle x2 + y2 - 4x - 6y - 12 = 0 is

a) 50π b) 100π c) 25π d) 10π150. The points A(3, 1), B(7, -1) and C(5, 0) are collinear.

Then the length of the tangent drawn from A to any circle

through B and C is

a) 10 b) √10

c) 2√10 d) None of these

151. The perimeter of the square, such that the circle (x - h)2 +

(y - k)2 = r2 is inscribed in the square is

a) 4r b) 8πr

c) 4πr d) 8r

152. The circles (x + a)2 + (y + b)2 = a2 and (x + α)2 + (y + β)2

= b2 intersect orthogonally, iff

a) aα + bα = α2 + β2 b) aα + bβ = a2 + b2

c) 2(aα + bβ) = α2 + β2 d) aα + bβ = b2 - a2

153. If the distance between the centres of two circle of radii

2sec2500 and 2tan2500 is 2, then they

a) touch externally b) touch internally

c) do not touch d) do not intersect

154. Which of the following lines is perpendicular to the line

of centres of the circles x2 + y2 + 2x + y + l = 0 and x2 + y2

- x + 2y + 3 = 0 ?

a) 3x - y = 0 b) 3x - y - 2 = 0

c) x + y = 0 d) none of these

155. The value of p for which the equation 7x2 + py2 - 5x + 7y

- 3 = 0 represents a circle is

a) 7 b) 6

c) 5 d) none of these

156.The circumference of the circle 4x2 + 4y2 - 4x - 4y = 7 is

a) 2π b) π c) 9π/4 d) 3π

RMA PBN

Pop

ade’

s R

elia

nce

Mat

hs A

cade

my

Par

bhan

i

157. If a circle passes through the point (a, b) and cuts the

circle x2 + y2 = 4 orthogonally, then the locus of its centre

is :

a) 2ax + 2by + {a2 + b2 + 4) = 0

b) 2ax + 2by - (a2 + b2 + 4) = 0

c) 2ax - 2by + (a2 + b2 + 4) = 0

d) 2ax - 2by - (a2 + b2 + 4) = 0

158. If the lines 2x + 3y + 1 = a and 3x - y - 4 = a lies along

diameter of a circle of circumference 10π, then the equation

of the circle is :

a) x2 + y2 - 2x + 2y - 23 = 0 b) x2 + y2 - 2x - 2y - 23 = 0

c) x2 + y2 + 2x + 2y - 23 = 0 d) x2 + y2 + 2x - 2y - 23 = 0

159. If the circles x2 + y2 + 2ax + cy + a = 0 and x2 + y2 - 3ax + dy

- 1 = 0 intersect in two distinct points P and Q, then the line

5x + by - a = 0 passes through P and Q for:

a) exactly two values of a b) infinitely many values of a

c) no value of a d) exactly one value of a

160. A circle touches the x-axis and also touches the circle

with centre at (0, 3) and radius 2. The locus of the centre

of the circle is :

a) a parabola b) a hyperbola

c) a circle d) an ellipse

161. Let AB be a chord of the circle x2 + y2 = r2 subtending a

right angle at the centre. Then the locus of the centroid of the

∆PAB as P moves on the circle is :

a) a parabola b) a circle

c) an ellipseC d) a pair of straight line

162. If a > 2b > 0, then the positive value of m for which the line

y = mx - b √l + m2 is a common tangent to x2 + y2 = b2

and (x - a)2 + y2 = b2, is :

a) b)

c) d)

163. If the tangent at the point P on the x2 + y2 + 6x + 6y = 2

meets the straight 5x - 2y + 6 = 0 at a point Q on the y-

axis, then the length of PQ is

a) 4 b) 2√5

c) 5 d) 3√5

164. The centre of circle inscribed in square formed by the

lines x2 - 8x + 12 = 0 and y2 -14y + 45 = 0, is

a) (4, 7) b) (7, 4)

c) (9, 4) d) (4, 9)

165. If one of the diameter of the x2 + y2 - 2x - 6y + 6 = 0 is a

chord to the circle with centre (2, 1), then the radius of

circle is :

a) √3 b) √2

c) 3 d) 2 .

166. The tangent at (1, 7) to the curve x2 = y - 6 touches the

circle x2 + y2 + 16x + 12y + c = 0 at :

a) (6, 7) b) (-6, 7)

c) (6, -7) d) (-6, -7)

167. A line through (0, 0) cuts the circle x2 + y2 - 2ax = 0 at A and

B, then locus of the centre of the circle drawn AB as

diameter is :,

a) x2 + y2 - 2ay = 0 b) x2 + y2 + ay = 0

c) x2 + y2 + ax = 0 d) x2 + y2 - ax = 0

168. Equation of radical axis of circles x2 + y2 - 3x - 4y + 5 = 0

and 2x2 + 2y2 - l0x - 12y + 12 = 0 is:

a) 2x + 2y -1 = 0 b) 2x + 2y + 1 = 0

c) x + y + 7 = 0 d) x + y - 7 = 0

2b

√a2 - 4b2

√a2 - 4b2

2b

2b

a - 2b

b

a - 2b

169. The equation of the circle which passes through the

intersection of x2 + y2 + 13x - 3y = 0 and 2x2 + 2y2 + 4x

-7y - 25 = 0 and whose centre lies on 13x + 30y = 0, is :

a) x2 + y2 + 30x -13y - 25 = 0

b) 4x2 + 4y2 + 30x -13y - 25 = 0

c) 2x2 + 2y2 + 30x -13y - 25 = 0

d) x2 + y2 + 30x -13y + 25 = 0

170. The number of common tangents that can be drawn to

two circles x2 + y2 = 6x and x2 + y2 + 6x + 2y + 1 = 0 is :

a) 4 b) 5

c) 2 . d) 1

171. The equation of the circle of radius 5 and touching the

co-ordinate axes in third quadrant is :

a) (x - 5)2 + (y + 5)2 = 25 b) (x + 4)2 + (y + 4)2 = 25

c) (x + 6)2 + (y + 6)2 = 25 d) (x + 5)2 + (y + 5)2 = 25

172. If P is a point such that the ratio of the squares of the lengths

of the tangents from P to the circles x2 + y2 + 2x - 4y - 20 = 0

and x2 + y2 - 4x + 2y - 44 = 0 is 2 : 3, then the locus of P is a

circle with centre:

a) (7, - 8) b) (-7, 8)

c) (7, 8) d) ( -7, - 8)

173. The circle x2 + y2 - 6x - 8y + 9 = 0 touches externally a

circle with centre at origin. Then the radius of the second

circle is

a) 1 b) 16

c) 21 d) none of these

174. The locus of the centre of the circle which cuts the circles

x2 + y2 + 2g1x + 2f

1y + c = 0 and x2 + y2 + 2g

2x + 2f

2y +

c2 = 0 orthogonally is

a) The radical axis of the given circles

b) a conic

c) another circle

d) none of these

175. One diameter of a circle is the join of A(-3, 7) and B(2, -5).

The tangent at A to the circle is

a) 5x - 12y + 99 = 0 b) 5x - 12y - 70 = 0

c) 5x - 12y + 44 = 0 d) none of these

176. The circle x2 + y2 = 9 meets the coordinate axes at A, B, C

and D. Then the area of the square ABCD is

a) 9 b) 36

c) 18 d) none of these .

177. From a point on the line x - y = 4, the lengths of the tan-

gents to the two circles x2 + y2 - 9 = 0 and x2 + y2 - 2x + 8y

- 7 = 0 are equal. Then the point is

a) (5, -2) b) (-5, 1)

c) (5, 1) d) (1, 5)

178. The line x - 3y = c meets the circle x2 + y2 = 2a2 at

exactly one point if c2 =

a) 40a2 b) 10a2

c) 20a2 d) none of these

179. The equation of the unit circle concentric with the circle

x2 + y2 - 4x + 6y - 12 = 0 is

a) x2 + yZ - 4x + 6y - 12 = 0 b) x2 + y2 - 4x + 6y - 15 = 0

c) x2 + y2 - 4x + 6y + 25 = 0 d) x2 + y2 - 4x + 6y + 12 = 0

180. The equations of the circles which touch x = 0, y = 0 and

x = c are

a) x2 + y2 - cx + cy + (1/4)c2 = 0

b) x2 + y2 + cx - cy + (1/4)c2 = 0

c) x2 + y2 - cx - cy + (1/4)c2 = 0

d) None of these

181.The number of tangents to the circle x2 + y2 - 8x - 6y + 9 = 0,

which pass through (3, -2) is

a) 1 b) 2

c) 0 d) None of these

182.The centre of the circle 16x2 + 16y2 + 160x - 256y -377 = 0

is

a) (-5, 8) b) (5, -8)

c) (-10, 16) d) none of these

RMA PBN

Pop

ade’

s R

elia

nce

Mat

hs A

cade

my

Par

bhan

i




a) 192 b) 192/24


135. The length of the chord joining the points (4cosθ, 4sinθ)

and (4cos(θ + 600), 4sin(θ + 600)) of the circle x2 + y2 = 16

a) 16 b) 2

c) 8 d) 4

136. Let PQ and RS be the tangents at the extrimities of the

diameter PR of a circle of radius r. If PS and QR intersect at

a point X on the circumference of the circle, then 2r equals

a) √PQ.RS b)

c) d)

PQ + RS

2

2PQ.RS

PQ + RSPQ2 + RS2

2

Pop

ade’

s R

elia

nce

Mat

hs A

cade

my

Par

bhan

i

183. The locus of the point whose shortest distance from the

circle x2 + y2 - 2x + 6y - 6 = 0 is equal to its distance from

the line x - 3 = 0

a) x2 + 6y - 4x - 9 = 0 b) y2 - 6y + 4x + 9 = 0

c) x2 - 6y - 4x - 9 = 0 d) y2 + 6y - 4x + 9 = 0

192. The circle coaxial with the circles x2 + y2 - 6x + 4y - 8 = 0

and x2 + y2 + 2x + y + 4 = 0 and passing through the origin is

a) x2 + y2 - 2x + 6y = 0 b) 3x2 + 3y2 - x + y = 0

c) 3x2 + 3y2 - 2x + 6y = 0 d) none of these

184. The equation of normal to the circle x2 + y2 = a2 at the

point (x1, y

1) of it is

a) xx1 + yy

1 = a2 b) xy

1 - yx

1 = 0

c) xy1 + yx

1 = 0 d) xy

1 - yx

1 = a2

185. The equation of normal to the circle x2 + y2 + 2gx + 2fy +

c = 0 at the point (x1, y

1) of it is

a) xx1 + yy

1 + g(x + x

1) + f(y + y

1) + c = 0

b) xx1 - yy

1 + g(x - x

1) + f(y - y

1) + c = 0

c) (y1 + f)x - (x

1 + g)y + (gy

1 - x

1f) = 0

d) (x1 + f)x - (y

1 + g)y + (gx

1 - y

1f) = 0

186. The equation of tangent to the circle x2 + y2 = 64 at the

point A(2π/3) of it is

a) x + 5y + 17 = 0 b) x - 3 y + 16 = 0

c) 5x + y - 17 = 0 d) 5x + y + 17 = 0

#187. The equation of common tangent to the two circles

x2 + y2 - 4x + 10y + 20 = 0 and x2 + y2 + 8x - 6y - 24 = 0 is

a) 3x - 4y - 11 = 0 b) 3x + 4y + 11 = 0

c) 4x - y - 13 = 0 d) x - 2y - 11 = 0

188. The equation of normal to the circle x2 + y2 = 25 at

(-4, -3) is

a) 3x - 4y = 0 b) 3x + 4y = 0

c) 4x + 3y + 25 = 0 d) 3x - 4y - 25 = 0

# Equation of common tangent can be obtained by substra-

cting one equation of circle from another equation.

189. The equation of normal to the circle 2x2 + 2y2 - 2x + 4y -

6 = 0 at the point (2, -5) is

a) 3x - 8y - 18 = 0 b) 3x - 8y = 0

c) 8x + 3y - 1 = 0 d) 8x + 3y - 25 = 0

190. If α and β are the inclinations of the tangents drawn to

the circle x2 + y2 = a2 from the point P, then the locus of

point P such that cotα.cotβ = 4 is

a) x2 + 4y2 - 3a2 = 0 b) 3x2 - 2xy - 3a2 = 0

c) 4x2 - 3y2 + a2 = 0 d) x2 - 4y2 + 3a2 = 0

191. The equation of director circle of the circle (x - 2)2 +

(y - 3)2 = 36 is

a) (x - 2)2 + (y - 3)2 = 72 b) (x + 2)2 + (y + 3)2 = 72

c) (x - 3)2 + (y - 2)2 = 72 d) (x + 3)2 + (y + 2)2 = 72

193. The equation of the circle which is concentric with the circle

x2 + y2 - 4x + 6y - 3 = 0 and having double its area is

a) x2 + y2 - 4x + 6y + 3 = 0 b) x2 + y2 - 4x + 6y - 51 = 0

c) x2 + y2 - 4x + 6y - 34 = 0 d) x2 + y2 - 4x + 6y - 19 = 0

194. The line 2x - y + 1 = 0 is a tangent to the circle at (2, 5) and

the centre lies on the line x + y = 9, then the equation of

circle is

a) x2 + y2 - 4x - 14y + 3 = 0 b) x2 + y2 - 12x - 6y - 20 = 0

c) x2 + y2 - 4x - 14y + 25 = 0 d) x2 + y2 - 12x - 6y + 25 = 0

195. The equation of the director circle of the circle x2 + y2 - 6x

+ 4y + 12 = 0 is

a) x2 + y2 - 6x + 4y + 11 = 0 b) x2 + y2 - 4x - 6y - 13 = 0

c) x2 + y2 - 6x + 4y + 15 = 0 d) x2 + y2 - 2x - 6y + 11 = 0

196. The tangents from the point P(3, 2) to the circle x2 + y2 - 6x

- 8y + 23 = 0 meet the circle at the points A and B. If C is

the centre of the circle, then PACB is a

a) parallelogram b) rectangle

c) rhombus d) Square

197. The locus of the point of intersection of the lines

xcosθ + ysinθ = a and xsinθ - ycosθ = b is a circle given by

a) x2 + y2 = a2 b) x2 + y2 = a2 - b2

c) x2 + y2 = a2 + b2 d) x2 + y2 = b2

198. The radii of the three circles x2 + y2 - 2x + 2y + 1 = 0,

x2 + y2 + 4x - 6y - 3 = 0 and x2 + y2 - 8x - 10y - 8 = 0 are

a) in G.P b) in A.P.

c) in H.P. d) equal

199. The centres of the three circles x2 + y2 + 4x + 12y + 2 = 0,

x2 + y2 - 2x - 6y - 9 = 0 and x2 + y2 = 16 form

a) an equilateral triangle b) an isosceles triangle

c) a right angled triangle d) are collinear

200. A and B are fixed points in a plane and a point P moves in

the plane such that PA = 2 PB, then the locus of P is

a) a circle b) a parabola

c) an ellipse d) a hyperbola

201. The equation of the diameter of the circle (x - 6)(x - 4) +

(y - 8)(y - 2) = 0, parallel to X-axis is

a) y = 5 b) y = 7

c) y + 5 = 0 d) y = 3

202. The equation of the circle circumscribing the triangle

formed by the lines y = x, y = 3x + 2 and y = 2x is

a) x2 + y2 - 3x + 4y = 0 b) x2 + y2 + 6x - 8y = 0

c) x2 + y2 + 6x + 8y = 0 d) x2 + y2 - 6x + 8y = 0

203. The equation of normal the circle x = acosθ, y = asinθ at

the point P(θ) of it is

a) xcosθ + ysinθ = a b) xcosθ - ysinθ = 0

c) xsinθ - ycosθ = 0 d) xsinθ - ycosθ = a

204. One of the lines given by the equation 6x2 - xy - y2 = 0

touches the circle x2 + y2 + 2x - 6y + 5 = 0, then the equa-

tion of that line is

a) 3x + y = 0 b) 2x - y = 0

c) 3x - y = 0 d) x - 2y = 0

205. Four distinct points (2k, 3k), (1, 0), (0, 1) and (0, 0) lie on

a circle for

a) For integral values of k b) 0 < k < 1

c) k < 0 d) k = 5/13

PQ + RS

2

2PQ.RS

PQ + RSPQ2 + RS2

2

206. The length of the chord joining the points (4cosθ, 4sinθ)

and (4cos(θ + 600), 4sin(θ + 600)) of the circle x2 + y2 = 16

a) 16 b) 2

c) 8 d) 4

207. Let PQ and RS be the tangents at the extrimities of the

diameter PR of a circle of radius r. If PS and QR intersect at

a point X on the circumference of the circle, then 2r equals

a) √PQ.RS b)

c) d)




a) 192 b) 192/24


209. The coordinates of point P(π/4) on the circle x2 + y2 = 25

a) (1/√2, 7/√2 ) b) (2, 3)

c) (5/√2, 5/√2 ) d) None of these

“A person’s probability of success is directly

proportional to the belief and execution of

their abilities.”

210. The coordinates of a point on the circle x2 + y2 = 9 whose

parameter is π/6

a) (3√3/2, 3/2) b) (3√3, 3/2)

c) (√3/2, √3/2) d) (3/2, 3/2)

211. (a, b) and (5, -1) are the end points of the diameter of the

circle x2 + y2 + 4x - 4y - 2 = 0, then the values of a, b resp.

a) 1, 3 b) 3, 1

c) -1, -3 d) None of these

RMA PBN

Documents

circl