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Pop
ade’
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Mat
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Par
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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’
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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
c) 12 d) None of these.
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
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ade’
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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
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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
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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
c) 12 d) None of these.
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
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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
c) 192/25 d) None of these
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
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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
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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
c) 192/25 d) None of these
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
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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)
208. 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
c) 192/25 d) None of these
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
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