Baixe 65878435 - Circles - amp - System - of - Circles - Qns e outras Exercícios em PDF para Matemática, somente na Docsity!
QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
1. If the line x + 2 by + 7 = 0, is a diameter of the circle, x^2 + y^2 - 6x + 2y = 0 then b = (A) 3 (B) - 5 (C) - 1 (D) 5 2. The equation of the circle which touches both the axes & whose radius is a, is : (A) x^2 + y^2 - 2 ax - 2 ay + a^2 = 0 (B) x^2 + y^2 + ax + ay = 0 (C) x^2 + y^2 + 2 ax + 2 by - a^2 = 0 (D) None of these 3. The centres of the circles, x^2 + y^2 = 1, x^2 + y^2 + 6x − 2y = 1 & x^2 + y^2 − 12x + 4y = 1 are : (A) Same (B) Collinear (C) Non−collinear (D) None of these 4. If a circle passes through the point (0, 0), (a, 0), (0, b), then its centre is (A) (a, b) (B) (b, a)
(C)
a b 2 2
(D)
b a 2 2
5. The line l x + my + n = 0 will be a tangent to the circle, x^2 + y^2 = a^2 if : (A) n^2 ( l^2 + m^2 ) = a^2 (B) a^2 ( l^2 + m^2 ) = n^2 (C) n ( l + m) = a (D) a ( l + m) = n 6. The angle between the two tangents from the origin to the circle, (x - 7)^2 + (y + 1)^2 = 25 is :
(A) 0 (B)
π 3
(C)
π 6
(D)
π 2
7. The locus of the middle points of
those chords of the circle, x^2 + y^2 = 4 which subtend a right angle at the origin is : (A) x^2 + y^2 - 2x - 2y = 0 (B) x^2 + y^2 = 4 (C) x^2 + y^2 = 2 (D) (x - 1)^2 + (y - 2)^2 = 5
8. A pair of tangents are drawn from the origin to the circle, x^2 + y^2 + 20 (x + y) + 20 = 0. The equation of the pair of tangents is : (A) x^2 + y^2 + 10 xy = 0 (B) x^2 + y^2 + 5 xy = 0 (C) 2 x^2 + 2 y^2 + 5 xy = 0 (D) 2 x^2 + 2 y^2 - 5 xy = 0 9. y = mx is a chord of a circle of radius a and the diameter of the circle lies along x-axis and one end of the chord is origin. The equation of the circle described on this chord as diameter is : (A) (1 + m^2 ) ( (x^2 + y^2 ) - 2 ax = 0 (B) (1 + m^2 ) (x^2 + y^2 ) − 2a (x + my) = 0 (C) (1 + m^2 ) (x^2 + y^2 ) + 2a (x + my) = 0 (D) (1 + m^2 ) (x^2 + y^2 ) − 2a (x − my) = 0 10. If two circles, (x − 1)^2 + (y − 3)^2 = r^2 and x^2 + y^2 - 8x + 2y + 8 = 0 intersect in two distinct points, then : (A) 2 < r < 8 (B) r = 2 (C) r < 2 (D) r > 2 11. The equation of the circle which touches both axes & whose centre is (x 1 , y 1 ) is : (A) x^2 + y^2 + 2x 1 (x + y) + x 12 = 0
(B) x^2 + y^2 − 2x 1 (x + y) + x 12 = 0
(C) x^2 + y^2 = x 12 + y 12 (D) x^2 + y^2 + 2 xx 1 + 2 yy 1 = 0
Circles & System of Circles
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
12. The lines, 2x − 3y = 5 & 3x − 4y = 7 are the diameters of a circle of area 154 sq. units. The equation of the circle is : (A) x^2 + y^2 + 2x - 2y = 62 (B) x^2 + y^2 - 2x + 2y = 47 (C) x^2 + y^2 + 2x - 2y = 47 (D) x^2 + y^2 - 2x + 2y = 62 13. A circle touches the y-axis at the point (0, 4) & cuts the x-axis in a chord of length 6 units. The radius of the circle is : (A) 3 (B) 4 (C) 5 (D) 6 14. If the line l x + my = 1 be a tangent to the circle x^2 + y^2 = a^2 , then the locus of the point ( l , m) is : (A) A straight line (B) A circle (C) A parabola (D) None of these 15. The equation of a circle which touches both axes and the line, 3x - 4y + 8 = 0 and lies in the third quadrant is : (A) x^2 + y^2 - 4x + 4y - 8 = 0 (B) x^2 + y^2 - 4x + 4y + 4 = 0 (C) x^2 + y^2 + 4x + 4y + 4 = 0 (D) x^2 + y^2 - 4x - 4y - 4 = 0 16. Tangents are drawn from the point (4, 3) to the circle x^2 + y^2 = 9. The area of the triangle formed by them and the line joining their points of contact is :
(A)
(B)
(C)
(D)
17. Circles x^2 + y^2 - 2x - 4y = 0 and
x^2 + y^2 - 8y - 4 = 0 , (A) Touch internally (B) Touch externally (C) Intersect each other at two distinct points (D) Do not intersect each other at any point
18. The equations to the tangents to the circle, x^2 + y^2 - 6x + 4y = 12 which are parallel to the straight line, 4x + 3y + 5 = 0, are : (A) 3x − 4y − 19 = 0, 3x − 4y + 31 = 0 (B) 4x + 3y − 19 = 0, 4x + 3y + 31 = 0 (C) 4x + 3y + 19 = 0, 4x + 3y − 31 = 0 (D) 3x − 4y + 19 = 0, 3x − 4y + 31 = 0 19. The length of tangent from the point (5, 1) to the circle, x^2 + y^2 + 6x - 4y - 3 = 0, is : (A) 81 (B) 29 (C) 7 (D) 21 20. The length of common chord of the circle, (x - a)^2 + y^2 = a^2 and x^2 + (y - b)^2 = b^2 is :
(A) (^2) a 2 + b^2 (B)
a b a 2 +b^2
(C)
2 2
a b a + b
(D) None of these
21. If y = 2x is a chord of the circle, x^2 + y^2 - 10x = 0, then the equation of the circle of which this chord is a diameter, is : (A) x^2 + y^2 - 2x + 4y = 0 (B) x^2 + y^2 + 2x + 4y = 0 (C) x^2 + y^2 + 2x - 4y = 0 (D) x^2 + y^2 - 2x - 4y = 0
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
passing through the points of intersection of the circles, x^2 + y^2 − 2x − 4y + 1 = 0 & x^2 + y^2 − 4x − 2y + 4 = 0 is : (A) x^2 + y^2 - 6x + 7 = 0 (B) x^2 + y^2 - 3y + 4 = 0 (C) x^2 + y^2 - 2x - 2y + 1 = 0 (D) x^2 + y^2 + 2x - 4y + 4 = 0
33. If a circle passes through the point (1, 2) and cuts the circle x^2 + y^2 = 4 orthogonally, then the equation of the locus of its centre is : (A) x^2 + y^2 - 3x - 8y + 1 = 0 (B) x^2 + y^2 - 2x - 6y - 7 = 0 (C) 2x + 4y - 9 = 0 (D) 2x + 4y - 1 = 0 34. A circle which passes through origin and cuts intercepts on axes a & b, the equation of the circle is : (A) x^2 + y^2 - ax - by = 0 (B) x^2 + y^2 + ax + by = 0 (C) x^2 + y^2 - ax + by = 0 (D) x^2 + y^2 + ax - by = 0 35. At which point on y-axis the line x = 0 is a tangent to circle, x^2 + y^2 - 2x - 6y = 0. (A) (0, 1) (B) (0, 2) (C) (0, 3) (D) (0, 4) 36. The equation of radical axis of the circles, x^2 + y^2 + x - y + 2 = 0 and 3x^2 + 3y^2 - 4x - 12 = 0, is : (A) 2x^2 + 2y^2 - 5x + y - 14 = 0 (B) 7x - 3y + 18 = 0 (C) 5x - y + 14 = 0 (D) None of these 37. The normal to the circle, x^2 + y^2 − 3x − 6y − 10 = 0 at the point (- 3, 4), is : (A) 2x + 9y - 30 = 0
(B) 9x - 2y + 35 = 0 (C) 2x - 9y + 30 = 0 (D) 2x - 9y - 30 = 0
38. The locus of centre of a circle passing through (p, q) and cuts orthogonally to circle, x^2 + y^2 = k^2 , is : (A) 2 px + 2 qy - (p^2 + q^2 + k^2 ) = 0 (B) 2 px + 2 qy - (p^2 - q^2 + k^2 ) = 0 (C) x^2 + y^2 − 3px − 4qy + (p^2 − q^2 − k^2 ) = 0 (D) x^2 + y^2 − 2px − 3qy + (p^2 − q^2 − k^2 ) = 0 39. Two tangents PQ and PR drawn to the circle, x^2 + y^2 − 2x − 4y − 20 = 0 from point P (16, 7). If the centre of the circle is C, then the area of quadrilateral PQCR will be : (A) 75 sq. units (B) 150 sq. units (C) 15 sq. units (D) None of these 40. At which point the line, y = x + 2 a touches to the circle, x^2 + y^2 = a^2
(A)
a a 2 2
(B) −^ −
a a 2 2
(C) ± ±
a a 2 2
, (D) ∓
a a 2 2
41. The co-ordinates of pole of line, l x + my + n = 0 with respect to circles x^2 + y^2 = 1, is :
(A)
n
m n
(B)^ −^ −
n
m n
(C)
n
m n
^
^
(D) −
^
n
m n
42. A point P moves in such a way that its ration of distances from two coplanar points is always fixed number (≠ 1). Then its locus is : (A) Straight line (B) Circle
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
(C) Parabola (D) A pair of straight lines
43. The two points A & B in a plane such that for all points P lies on circle
satisfied
PA
PB
= k, then k will not be
equal to : (A) 0 (B) 1 (C) 2 (D) None of these
44. The area of triangle formed by the
tangents, normal drawn at (^) ( 1 , 3 ) to the circle x^2 + y^2 = 4 and positive x−axis is :
(A) 2 3 (B) 3
(C) 4 3 (D) None of these
45. A circle passes through (0, 0) & (1, 0) and touches to the circle, x^2 + y^2 = 9, then the centre of circle is :
(A)
^
^
(B)
^
(C)
^
^
(D)
^
46. From the origin, chords are drawn to the circle, (x - 1)^2 + y^2 = 1. The locus of mid points of these chords is a : (A) Circle (B) Straight line (C) Pair of straight line (D) None of these 47. The tangents are drawn from the point (4, 5) to the circle, x^2 + y^2 − 4x − 2y − 11 = 0. The area of quadrilateral formed by these tangents and radii is (A) 15 sq. units (B) 75 sq. units (C) 8 sq. units (D) 4 sq. units 48. A chord AB drawn from the point
A (0, 3) at circle x^2 + 4x + (y - 3)^2 = 0 and it meets to M in such a way that AM = 2 AB, then the locus of point M will be : (A) Straight line (B) Circle (C) Parabola (D) None of these
49. The co-ordinates of the point from where the tangents are drawn to the circles x^2 + y^2 = 1, x^2 + y^2 + 8x + 15 = 0 and x^2 + y^2 + 10y + 24 = 0 are of same length are :
(A) ( 2 , 52 ) (B) ( − 2 ,−^52 )
(C) ( − 2 , 52 ) (D) ( 2 , −^52 )
50. The equation of the circumcircle of the triangle formed by the lines,
y + 3 x = 6, y - 3 x = 6 & y = 0, is (A) x^2 + y^2 - 4y = 0 (B) x^2 + y^2 + 4x = 0 (C) x^2 + y^2 - 4y = 12 (D) x^2 + y^2 + 4x = 12
51. If the points (2, 0), (0, 1), (4, 5) and (0, c) are concyclic, then c is equal to
(A) - 1, -
(B) - 1, -
(C) 14
, 1 (D) None of these
52. The abscissae of A and B are the roots of the equation, x^2 + 2 ax - b^2 = 0 and their ordinates are the roots of the equation, y^2 + 2 py - q^2 = 0. The equation of the circle with AB as diameter, (A) x^2 + y^2 + 2 ax + 2 py - b^2 - q^2 = 0 (B) x^2 + y^2 + 2 ax + py - b^2 - q^2 = 0 (C) x^2 + y^2 + 2 ax + 2 py + b^2 + q^2 = 0
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
(C) The incentre of the triangle (D) The centroid
65. Consider the circles x^2 + (y − 1)^2 = 9, (x − 1)^2 + y^2 = 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 66. The locus of centre of the circle which cuts the circles, x^2 + y^2 + 2 g 1 x + 2 f 1 y + c 1 = 0 orthogonally is : (A) An ellipse (B) The radical axis of the given circles (C) A conic (D) Another circle 66. Locus of a point which moves such that sum of the squares of its distances from the sides of a square of side unity is 9, is : (A) Straight line (B) Circle (C) Parabola (D) None of these 67. Polar of origin (0, 0) w.r.t. the circle, x^2 + y^2 + 2 λx + 2 μy + c = 0 touches circle x^2 + y^2 = r^2 , if : (A) c = r (λ^2 + μ^2 ) (B) r = c (λ^2 + μ^2 ) (C) c^2 = r^2 (λ^2 + μ^2 ) (D) r^2 = c^2 (λ^2 + μ^2 ) 68. Line, Ax + By + C = 0 cuts circle, x^2 + y^2 + ax + by + c = 0 in P & Q and the line A ′x + B ′y + C ′ = 0 cuts the circle x^2 + y^2 + a ′x + b ′y + c ′ = 0 in R & S. If the four points P, Q, R & S are concyclic, then
D =
a a b b c c A B C A B C
(A) 1 (B) 0
(C) - 1 (D) None of these
69. Circles, (x + a)^2 + (y + b)^2 = a^2 and (x + α)^2 + (y + β)^2 = β^2 cuts orthogonally if : (A) a α + b β = b^2 + a^2 (B) 2 (a α + b β) = b^2 + a^2 (C) a α + b β = a^2 + b^2 (D) None of these 70. If the circles of same radius a and centres at (2, 3) and (5, 6) cut orthogonally, then a = (A) 1 (B) 2 (C) 3 (D) 4 71. A circle is inscribed in an equilateral triangle of side a, the area of any square inscribed in the circle is :
(A)
a 2 3
(B)
a^2
(C)
a^2 6
(D)
a^2 12
72. The angle between a pair of tangents drawn from a point P to the circle, x^2 + y^2 + 4x − 6y + 9 sin^2 α + 13 cos^2 α = 0 is 2 α. The equation of the locus of the point P is : (A) x^2 + y^2 + 4x - 6y + 4 = 0 (B) x^2 + y^2 + 4x - 6y - 9 = 0 (C) x^2 + y^2 + 4x - 6y - 4 = 0 (D) x^2 + y^2 + 4x - 6y + 9 = 0 73. The intercept on the line y = x, by the circle, x^2 + y^2 − 2x = 0 is AB.
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
Equation of the circle with AB as a diameter is : (A) x^2 + y^2 - x - y = 0 (B) x^2 + y^2 - 2x - y = 0 (C) x^2 + y^2 - x + y = 0 (D) x^2 + y^2 + x - y = 0
74. The circles, x^2 + y^2 + 4x + 6y + 3 = 0 and 2 (x^2 + y^2 ) + 6x + 4y + C = 0 will cut orthogonally, if C equals : (A) 4 (B) 18 (C) 12 (D) 16
ANSWERS
1. D 2. A 3. B 4. C 5. B 6. D
7. C 8. C 9. B 10. C 11. B 12.B
13. C 14. B 15. C 16. C 17. A 18.C
19. C 20. C 21. A 22. D 23. A 24.D
25. A 26. A 27. A 28. D 29. D 30.C
31. C 32. A 33. C 34. A 35. C 36.B
37. A 38. A 39. A 40. D 41. B 42.B
43. B 44. A 45. D 46. A 47. C 48.B
49. B 50. C 51. C 52. A 53. A 54.B
55. A 56. D 57. D 58. A 59. B 60.A
61. D 62. C 63. A 64. A 65. B 66.B
67. C 68. B 69. B 70. C 71. C 72.D
73. A 74. D
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