CIRCLES

1. The number of common tangents to the two circles x2 + y2 = 4 and x2 + y2 – 8x + 12 = 0 is

(A) 1 (B) 2 (C) 3 (D) 4

2. The distance of the point (1, 2) from the common chord of the circles x 2 + y2 – 2x + 3y – 5 = 0
and x2 + y2 + 10x + 8y – 1 = 0 is _______________

3. The equation of the circle passing through the origin and the points of intersection of the two
circles x2 + y2 – 4x – 6y – 3 = 0 and x2 + y2 + 4x – 2y – 4 = 0 is ___________
4. The equation of the units circle concentric with x2 + y2 – 8x + 4y – 8 = 0 is
(A) x2 + y2 – 8x + 4y – 8 = 0 (B) x2 + y2 – 8x + 4y + 8 = 0
(C) x2 + y2 – 8x + 4y – 28 = 0 (D) x2 + y2 – 8x + 4y + 19 = 0

5. Circles x2 + y2 – 4x – 6y – 12 = 0 and x2 + y2 + 4x + 6y + 4 = 0
(A) touch externally (B) internally
(C) intersect at two point (D) do not intersect

6. The locus of the middle point of the chord of the circle x 2 + y2 – 2x = 0 passing through the
origin is _____________
7. The radius of any circle touching the lines 3x – 4y + 5 = 0 and 6x – 8y – 9 = 0 is __________

1
8. If f is from (–, 0) into R and f(x) = , then f(x) = _______________
x

9. A circle touches the x-axis and also touches the circle with centre (0, 3) and radius 2. The
locus of the centre of the circle is
(A) a circle (B) a parabola (C) an ellipse (D) a hyperbola

10. The equation of the circle described on the common chord of the circles x 2 + y2 + 2x = 0 and
x2 + y2 + 2y = 0 as diameters is
(A) x2 + y2 + x – y = 0 (B) x2 + y2 – x – y = 0
(C) x + y – x + y = 0
2 2 (D) x2 + y2 – x – y = 0

11. Origin is a limiting point of a coaxal of which x 2 + y2 – 6x – 8y + 1 = 0 is a member. The other

limiting point is
(A) (–2, –4) (B) (3/25, 4/25) (C) (–3/25, –4/25) (D) (4/25, 3/25)

12. A circle passes through the origin and has its centre on y = x. if it cuts
x2 + y2 – 4x – 6y + 10 = 0 orthogonally, its equation is
(A) x2 + y2 – x – y = 0 (B) x2 + y2 – 6x + 4y = 0
(C) x + y – 2x – 2y = 0
2 2 (D) x2 + y2 + 2x + 2y = 0

13. The radical centre of the three circles described on the three sides of a triangle as diameter is
(A) the orthocenter (B) the circumcentre
(C) the incrntre (D) the centroid of the triangle

14. The number of common tangents to the circles x2 + y2 – x = 0, x2 + y2 + x = 0 is

(A) 2 (B) 1 (C) 4 (D) 3

15. Consider the circles x2 + (y – 1)2 = 9, (x – 1)2 + y2 = 25, they are such that
(A) these circles touch each other
(B) one of these circles lies entirely inside the other
(C) each of these circles lies outside the other
(D) they intersect in two points.

16. The length of tangent from (0, 0) to the circle 2(x 2 + y2) + x – y + 5 = 0 is

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 5 5
(A) 5 (B) (C) 2 (D)
2 2

17. The circumcentre of the triangle formed by the lines xy + 2x + 2y + 4 = 0 and x + y + 2 = 0 is

(A) (0, 0) (B) (–2, –2) (C) (–1, –1) (D) (–1, –2)

18. The centre and the radius of the circle with the segment of the line x + y = 1 cut off by the co-
ordination axes as a diameter
 1 1  1 1 1
(A) (1, 1) 2 (B)  ,  , 2 (C)  ,  , (D) (0, 0), 1
 2 2  2 2 2

19. The equation of the circle which touches the lines x = 0, y = 0 and x = c is
(A) x2 + y2 – cx – cy + c2 = 0 (B) x2 + y2 – 2cx – 2cy + c2 = 0
2
c c2
(C) x2 + y2 + cx + cy + =0 (D) x2 + y2 – cx – cy + =0
4 4

20. The points (4, –2), (3, b) are conjugate with respect to the circle x2 + y2 = 24, if b =
(A) 6 (B) –6 (C) 12 (D) –4

21. A circle of the co-axial system with limiting points (0, 0) and (1, 0) is
(A) x2 + y2 – 2x = 0 (B) x2 + y2 – 6x + 3 = 0
(C) x2 + y2 = 1 (D) x2 + y2 – 2x + 1 = 0

22. If a polar of a point on the circle x2 + y2 = a2, with respect to the x2 + y2 = b2, touches the circle
x2 + y2 = c2, then a, b, c are
(A) A.P. (B) H.P. (C) G.P. (D) none of these

23. From the origin, chords are drawn to the circle x 2 + y2 – 2y = 0. The locus of the middle point
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

24. If the circles of same radius ‘a’ and centres at (2, 3) and (5, 6) cut orthogonally, then a =
(A) 3 (B) 4 (C) 6 (D) 10

25. The radius of the circle which has the lines x + y – 1 = 0 and x + y – 9 = 0 as tangents is
(A) 2 (B) 22 (C) 32 (D) 42

26. The polar of (–2, 3) with respect to x2 + y2 – 4x – 6y + 5 = 0 is

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

27. Equation of the tangent at (1, 1) to the circle 2x2 + 2y2 – 2x – 5y + 3 = 0 is

(A) 2x + y – 1 = 0 (B) 2x – y – 1 = 0 (C) x + 2y – 1 = 0 (D) 2x + y + 1 = 0

28. Given the circle x2 + y2 = 25, the equation of its chord with (1, –1) as the mid-point
(A) x + y = 2 (B) x + y + 2 = 0 (C) x – y = 2 (D) 2x – y = 2

29. If (0, 0) is one limiting point of a coaxial system of circles whose common radical axis is the
line x + y = 1, then the other limiting point is
(A) (1, 1) (B) (–1, –1) (C) (1, –1) (D) (–1, 1)

30. Given that for the circle x2 + y2 – 4x + 6y + 1 = 0 the line with equation 3x – y = 1 is a chord.
The middle point of the chord is
(A) (2/5, 11/5) (B) (–2/5, 11/5) (C) (–2/5, –11/5) (D) (2/5, –11/5)

31. The radical axis of the coaxial system of circles with limiting points (1, 2) and (4, 3) is given by
the equation
(A) 3x – y + 10 = 0 (B) 3x + y – 10 = 0 (C) 3x + y + 10 = 0 (D) x + 3y – 10 = 0
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32. The equation of a circle with centre (4, 1) and having 3x + 4y – 1 = 0 as tangent is
(A) x2 + y2 – 8x – 2y – 8 = 0 (B) x2 + y2 – 8x – 2y + 8 = 0
(C) x2 + y2 – 8x + 2y + 8 = 0 (D) x2 + y2 – 8x – 2y + 4 = 0

33. The pole of the line 8x – 2y = 11 with respect to the circle 2x2 + 2y2 = 11 is
(A) (4, 1) (B) (4, –1) (C) (3, 1) (D) (4, 2)

34. The radical axis of the circle x2 + y2 – 6x – 4y – 44 = 0 and x2 + y2 – 14x – 5y – 24 = 0 is

(A) 8x + y – 30 = 0 (B) 8x + y + 20 = 0 (C) 8x + 3y – 20 = 0 (D) 8x + y – 20 = 0

35. If (1, 2), (4, 3) are the limiting points of a system of coaxial circles then the radical axis of the
system is
(A) 3x + 2y – 10 = 0 (B) 3x + y – 10 = 0 (C) 2x + y = 10 (D) x + y – 6 = 0

36. The length of the tangent of from (6, 8) to the circle x2 + y2 = 4 is

(A) 6 (B) 2 6 (C) 4 6 (D) 5 6

37. The equation to the normal to the circle x2 + y2 – 2x – 2y – 2 = 0 is at the point (3, 1) on it is
(A) x = 1 (B) y = 2 (C) y = 1 (D) y = –1

38. A = (–9, 0) and B = (–1, 0) are two points. If P = (x, y) is a point such that 3PB = PA, then the
locus P is
(A) x2 – y2 = 9 (B) x2 – y2 = –9 (C) x2 + y2 = 9 (D) x2 + y2 = 3

39. The locus of point of intersection of perpendicular tangents to the circle x 2 + y2 = a2 is

(A) x2 + y2 = 2a2 (B) x2 + y2 = 4a2 (C) x2 + y2 = 6a2 (D) x2 + y2 = 8a2

2a(1 − t 2 ) 4at
40. The parametric equations x = and y = represent a circle of radius
1+ t 2
1 + t2
(A) a/2 (B) a (C) 2a (D) 4a

41. The inverse point (1, –1) with respect to the circle x2 + y2 = 4 is
(A) (–1, 1) (B) (–2, 2) (C) (1, –1) (D) (2, –2)

42. The equation to the polar of (–2, 3) with respected to x2 + y2 – 4x – 6y + 5 = 0 is

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

43. If the circle x2 + y2 + 2x – 2y + 4 = 0 cuts the circle x2 + y2 + 4x + 2fy + 2 = 0 orthogonally,

then f =
(A) 1 (B) 2 (C) –1 (D) –2

44. The slope of the radical axis of the circles x2 + y2 + 3x + 4y – 5 = 0 and

x2 + y2 – 5x – 5y – 6 = 0 is
(A) 1 (B) 3 (C) 5 (D) 8

45. The centre of the circle touching the y-axis at (0, 3) and making an intercept of 2 units on the
positive x-axis is
(A) (10, 3) (B) (3, 10) (C) (10, 3) (D) (3, 10)

46. The slope m of a tangent through the point (7, 1) to the circle x 2 + y2 = 25 satisfies the
equation
(A) 12m2 + 7m – 12 = 0 (B) 16m2 – 24m + 9 = 0
(C) 12m2 – 7m – 12 = 0 (D) 9m2 + 24m + 16 = 0

47. Two circles of equal radius ‘r’ cut orthogonally. If their centre are (2, 3) and (5, 6), then r =
(A) 1 (B) 2 (C) 3 (D) 4

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48. The number of common tangents that can be drawn to the circles x 2 + y2 = 1 and
x2 + y2 – 2x – 6y + 6 = 0 is
(A) 1 (B) 2 (C) 3 (D) 4

49. If (1, 2) is a limiting point of a coaxal system of circles containing the circle
x2 + y2 + x – 5y + 9 = 0, then the equation of radical axis is
(A) x + 3y + 9 = 0 (B) 3x – y + 4 = 0 (C) x + 0y –4 = 0 (D) 3x – y – 1 = 0

50. A variable circle passes through the fixed point (2, 0) and touches the y-axis. Then the locus
of its centre is
(A) a parabola (B) a circle (C) an ellilpse (D) a hyperbola

51. The radius of the circle r2 – 22 r(cos  + sin ) – 5 = 0 is

(A) 9 (B) 5 (C) 3 (D) 2

52. The equation of the normal to the circle x2 + y2 + 6x + 4y – 3 = 0 at (1, –2) is

(A) y + 1 = 0 (B) y + 2 = 0 (C) y + 3 = 0 (D) y – 2 = 0

53. The limiting points of the coaxal system containing the two circles x2 + y2 + 2x – 2y + 2 = 0
and 25(x2 + y2) – 10x – 80y + 65 = 0 are:
 −1 −8  1 8  −1 −8 
(A) (1, –1)(–5, –40) (B) (1, –1),  ,  (C) (–1, 1),  ,  (D) (–1, 1),  , 
 5 5   5 5  5 5 

54. The radical axis of the circles x2 + y2 + 3x + 4y – 5 = 0 and x2 + y2 – 5x + 5y – 6 = 0 is

(A) 8y – x + 1 = 0 (B) 8x – y + 1 = 0 (C) 8x – 8y + 1 = 0 (D) y – 8x + 1 = 0

55. If the polar of a point on the circle x 2 + y2 = p2 with respect to the circle x2 + y2 = q2 touches
the circle x2 + y2 = r2, then p, q, r are in ....... progression.
(A) Arithmetic (B) Geometric (C) Harmonic (D) Arithmetico Geometric

56. The equation of the circle of radius 5 and touching the coordinate axes in third quadrant is
(A) (x – 5)2 + (y + 5)2 = 25 (B) (x + 5)2 + (y + 5)2 = 25
2 2
(C) (x + 4) + (y + 4) = 25 (D) (x + 6)2 + (y + 6)2 = 25

57. The radius of the larger circle lying the first quadrant and touching the line 4x + 3y – 12 = 0
and the coordinate axes is
(A) 5 (B) 6 (C) 7 (D) 8

58. The four distinct point (0, 0), (2, 0), (0, –2) and (k, –2) are concylic if k =
(A) 2 (B) –2 (C) 0 (D) 1

59. A line is at a constant distance C from the origin and meets the coordinate axes in A and B.
The locus of the centre of the circle passing through O, A, B is
(A) x–2 + y–2 = c–2 (B) x–2 + y–2 = 2c–2 (C) x–2 + y–2 = 3c–2 (D) x–2 + y–2 = 4c–2

60. A variable circle passes through the fixed point (2, 0) and touches the Y-axis. Then locus of
its centre is
(A) A parabola (B) A circle (C) An ellipse (D) A hyperbola
61. If the circle x2 + y2 + 6x – 2y + k = 0 bisects the circumference of the circle
x2 + y2 + 2x – 6y – 15 = 0, then k = ...........
(A) 21 (B) –21 (C) 23 (D) –23

62. 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)

63. If 5x – 12y + 10 = 0 and 12y – 5x + 16 = 0 are two tangents to a circle, then the radius of the
circle is:
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 (A) 1 (B) 2 (C) 4 (D) 6

64. The number of circles that touch all the three lines x + y – 1 = 0, x – y – 1 = 0 and y + 1 = 0 is:
(A) 2 (B) 3 (C) 4 (D) 1

65. If P1, P2, P3 are the perimeters of the three circles x2 + y2 + 8x – 6y = 0, 4x2 + 4y2 – 4x – 12y –
186 = 0 and x2 + y2 – 6x + 6y – 9 = 0 respectively, then:
(A) P1 < P2 < P3 (B) P1 < P3 < P2 (C) P3 < P2 < P1 (D) P2 < P3 < P1

66. If the line 3x – 2y + 6 = 0 meets X-axis and Y-axis respectively at A and B, then the equation
of the circle with radius AB and centre at A is
(A) x2 + y2 + 4x + 9 = 0 (B) x2 + y2 + 4x – 9 = 0
(C) x2 + y2 + 4x + 4 = 0 (D) x2 + y2 + 4x – 4 = 0

67. A line l meets the circle x2 + y2 = 61 in A, B and P (–5, 6) is such that PA = PB = 10. Then the
equation of l is :
(A) 5x + 6y + 11 = 0 (B) 5x – 6y – 11 = 0 (C) 5x – 6y + 11 = 0 (D) 5x – 6y + 12 = 0

68. If (1, a), (b, 2) are conjugate points with respect to the circle x 2 + y2 = 25, then 4a + 2b =
(A) 25 (B) 50 (C) 100 (D) 150

69. The centre of the circle r2 – 4r (cos  + sin ) – 4 = 0 in Cartesian coordinates is:
(A) (1, 1) (B) (–1, –1) (C) (2, 2) (D) (–2, –2)

70. The radius of the circle r = 3 sin  + cos  is:

(A) 1 (B) 2 (C) 3 (D) 4

71. The equation of the ci9rcle whose diameter is the common chord of the circles
x2 + y2 + 2x + 3y + 2 = 0 and x2 + y2 + 2x – 3y – 4 = 0 is:
(A) x2 + y2 + 2x + 2y + 2 = 0 (B) x2 + y2 + 2x + 2y – 1 = 0
(C) x2 + y2 + 2x + 2y + 1 = 0 (D) x2 + y2 + 2x + 2y + 3 = 0

72. If x – y + 1 = 0 meets the circle x 2 + y2 + y – 1 = 0 at A and B, then the equation of the circle
with AB as diameter is:
(A) 2(x2 + y2) + 3x – y + 1 = 0 (B) 2(x2 + y2) + 3x – y + 2 = 0
(C) 2(x + y ) + 3x – y + 3 = 0
2 2 (D) x2 + y2 + 3x – y + 1 = 0

73. If y = 3x is a tangent to a circle with centre (1, 1), then the other tangent drawn through (0, 0)
to the circle is:
(A) 3y = x (B) y = –3x (C) y = 2x (D) y = –2x

74. Which of the following equations gives a circle?

(A) r = 2 sin  (B) r2 cos 2 = 1
(C) r(4 cos  + 5 sin ) = 3 (D) 5 = r(1 + 2 cos )

75. The number of common tangents to the two circles x 2 + y2 – 8x + 2y = 0 and x2 + y2 – 2x –

16y + 25 = 0 is:
(A) 1 (B) 2 (C) 3 (D) 4

76. Observe the following statements:

I. The circle x2 + y2 – 6x – 4y – 7 = 0 touches y-axis
II. The circle x2 + y2 + 6x + 4y – 7 = 0 touches x-axis
Which of the following is a correct statement?
(A) Both I and II are true (B) Neither I nor II is true
(C) I is true, II is false (D) I is false, II is true

77. The length of the tangent drawn to the circle x2 + y2 – 2x + 4y – 11 = 0 from the point (1, 3) is:
(A) 1 (B) 2 (C) 3 (D) 4

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  
78. The polar equation of the circle with centre  2,  and radius 3 units is:
 2
(A) r2 + 4r cos  = 5 (B) r2 + 4r sin  = 5 (C) r2 – 4r sin  = 5 (D) r2 – 4r cos  = 5

x2 y2
79. The radius of the circle passing through the foci of the ellipse + = 1 and having its centre
16 9
at (0, 3) is
(A) 4 (B) 72 (C) 3 (D) 12

80. The equation of the tangent to the circle x 2 + y2 + 4x – 4y + 4 = 0 which make equal intercepts
on the positive coordinate axes is
(A) x + y = 2 (B) x + y = 2 2 (C) x + y = 4 (D) x + y = 8

81. If the two circles (x – 1)2 + (y – 3)2 = r2 and x2 + y2 – 8x + 2y + 8 = 0. Intersect in two district
points, then
(A) r < 2 (B) r = 2 (C) r > 2 (D) 2 < r < 8

82. The lines 2x – 3y = 5 and 3x – 4y = 7 are diameters of a circle having area as 154 sq. units.
Then the equation of the circle is
(A) x2 + y2 + 2x – 2y = 47 (B) x2 + y2 – 2x + 2y = 47
(C) x + y – 2x + 2y = 62
2 2 (D) x2 + y2 + 2x – 2y = 62

83. If x1, x2, x3 and y1, y2, y3 are both in G.P. with the same common ratio, then the points
(x1, y1), (x2, y2) and (x3, y3)
(A) lie on an ellipse (B) lie on circle
(C) are vertices of triangle (D) lie on a straight line

84. 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

85. 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 = 4qx (B) (x – q)2 = 4py (C) (x – p)2 = 4qy (D) (y – q)2 = 4px

86. If the lines 2x + 3y + 1 = 0 and 3x – y – 4 = 0 lie along diameters of a circle of circumference

10, 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

87. The intercept on the line y = x by the circle x 2 + y2 – 2x = 0 is AB. Equation of the circle on AB
as a diameter is
(A) x2 + y2 + x + y = 0 (B) x2 + y2 – x – y = 0 (C) x2 + y2 – x – y = 0 (D) x2 + y2 + x – y = 0

88. If the circles x2 + y2 + 2ax + cy + a = 0 and x2 + y2 – 3ax + dy – 1 = 0 interesect in two distinct

points P and Q then the line 5x + by – a = 0 passes through P and Q for
(A) exactly one value of a (B) exactly two values of a
(C) infinitely many values of a (D) no value of a

89. A circle touches 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 circle (B) a hyperbola (C) an ellipse (D) a parabola

90. If a circle passes through the point (a, b) and cuts the circle x 2 + y2 = p2 orthogonally, then the
equation of the locus of its centre is
(A) 2ax + 2by – (a2 + b2 + p2) = 0 (B) 2ax + 2by – (a2 – b2 + p2) = 0
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 (C) x2 + y2 – 3ax – 4by + (a2 + b2 – p2) = 0 (D) x2 + y2 – 2ax – 3by + (a2 – b2 – p2) = 0

91. Let C be the circle with centre (0, 0) and radius 3 units. The equation of the locus of the mid
2
points of the chords of the circle C that subtends an angle of at its centre is
3
3 27 9
(A) x2 + y2 = (B) x2 + y2 = 1 (C) x2 + y2 = (D) x2 + y2 =
2 4 4

ANSWERS (CIRCLES)

1. C 2. 2 3. x2 + y2 – 28x – 18y = 0 4. D

5. D 6. x2 + y2 – x = 0 7. 19/20 8. f(x) = log |x| + C

9. B 10. D 11. B 12. C

13. B 14. D 15. B 16. D

17. C 18. C 19. D 20. B

21. D 22. C 23. A 24. A

25. B 26. A 27. B 28. C

29. A 30. C 31. B 32. B

33. B 34. D 35. B 36. C

37. C 38. C 39. A 40. C

41. D 42. C 43. C 44. D

45. C 46. C 47. C 48. D

49. B 50. A 51. C 52. B

53. C 54. B 55. B 56. B

57. B 58. A 59. D 60. A

61. D 62. B 63. A 64. C

65. B 66. B 67. C 68. B

69. C 70. A 71. C 72. A

73. A 74. A 75. B 76. B

77. C 78. C 79. A 80. B

81. D 82. B 83. D 84. B

85. C 86. C 87. C 88. D

89. D 90. A 91. D

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