Baixe 65878435 - Circles - amp - System - of - Circles - Qns e outras Exercícios em PDF para Matemática, somente na Docsity! 1 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, x2 + y2 - 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) x2 + y2 - 2 ax - 2 ay + a2 = 0 (B) x2 + y2 + ax + ay = 0 (C) x2 + y2 + 2 ax + 2 by - a2 = 0 (D) None of these 3. The centres of the circles, x2 + y2 = 1, x2 + y2 + 6x − 2y = 1 & x2+ y2 − 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 lx + my + n = 0 will be a tangent to the circle, x2 + y2 = a2 if : (A) n2 (l2 + m2) = a2 (B) a2 (l2 + m2) = n2 (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, x2 + y2 = 4 which subtend a right angle at the origin is : (A) x2 + y2 - 2x - 2y = 0 (B) x2 + y2 = 4 (C) x2 + y2 = 2 (D) (x - 1)2 + (y - 2)2 = 5 8. A pair of tangents are drawn from the origin to the circle, x2 + y2 + 20 (x + y) + 20 = 0 . The equation of the pair of tangents is : (A) x2 + y2 + 10 xy = 0 (B) x2 + y2 + 5 xy = 0 (C) 2 x2 + 2 y2 + 5 xy = 0 (D) 2 x2 + 2 y2 - 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 + m2) ( (x2 + y2) - 2 ax = 0 (B) (1 + m2) (x2 + y2) − 2a (x + my) = 0 (C) (1 + m2) (x2 + y2) + 2a (x + my) = 0 (D) (1 + m2) (x2 + y2) − 2a (x − my) = 0 10. If two circles, (x − 1)2 + (y − 3)2 = r2 and x2 + y2 - 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) x2 + y2 + 2x 1 (x + y) + x 1 2 = 0 (B) x2 + y2 − 2x 1 (x + y) + x 1 2 = 0 (C) x2 + y2 = x 1 2 + y 1 2 (D) x2 + y2 + 2 xx 1 + 2 yy 1 = 0 Circles & System of Circles www.myengg.com The Engineering Universe 1 Entrance Exams ,Engineering colleges in india, Placement details of IITs and NITs ww w. my en gg .co m 2 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) x2 + y2 + 2x - 2y = 62 (B) x2 + y2 - 2x + 2y = 47 (C) x2 + y2 + 2x - 2y = 47 (D) x2 + y2 - 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 lx + my = 1 be a tangent to the circle x2 + y2 = a2, 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) x2 + y2 - 4x + 4y - 8 = 0 (B) x2 + y2 - 4x + 4y + 4 = 0 (C) x2 + y2 + 4x + 4y + 4 = 0 (D) x2 + y2 - 4x - 4y - 4 = 0 16. Tangents are drawn from the point (4, 3) to the circle x2 + y2 = 9 . The area of the triangle formed by them and the line joining their points of contact is : (A) 24 25 (B) 64 25 (C) 192 25 (D) 192 5 17. Circles x2 + y2 - 2x - 4y = 0 and x2 + y2 - 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, x2 + y2 - 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, x2 + y2 + 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 + y2 = a2 and x2 + (y - b)2 = b2 is : (A) 2 a b2 2+ (B) a b a b 2 2+ (C) 2 2 2 a b a b+ (D) None of these 21. If y = 2x is a chord of the circle, x2 + y2 - 10x = 0, then the equation of the circle of which this chord is a diameter, is : (A) x2 + y2 - 2x + 4y = 0 (B) x2 + y2 + 2x + 4y = 0 (C) x2 + y2 + 2x - 4y = 0 (D) x2 + y2 - 2x - 4y = 0 www.myengg.com The Engineering Universe 2 Entrance Exams ,Engineering colleges in india, Placement details of IITs and NITs ww w. my en gg .co m 5 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 x2 + y2 = 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, x2 + y2 = 9, then the centre of circle is : (A) 3 2 1 2 ,       (B) 1 2 3 2 ,       (C) 1 2 1 2 ,       (D) 1 2 2, ±       46. From the origin, chords are drawn to the circle, (x - 1)2 + y2 = 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, x2 + y2 − 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 x2 + 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 x2 + y2 = 1, x2 + y2 + 8x + 15 = 0 and x2 + y2 + 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) x2 + y2 - 4y = 0 (B) x2 + y2 + 4x = 0 (C) x2 + y2 - 4y = 12 (D) x2 + y2 + 4x = 12 51. If the points (2, 0), (0, 1), (4, 5) and (0, c) are concyclic, then c is equal to (A) - 1, - 3 14 (B) - 1, - 14 3 (C) 14 3 , 1 (D) None of these 52. The abscissae of A and B are the roots of the equation, x2 + 2 ax - b2 = 0 and their ordinates are the roots of the equation, y2 + 2 py - q2 = 0 . The equation of the circle with AB as diameter, (A) x2 + y2 + 2 ax + 2 py - b2 - q2 = 0 (B) x2 + y2 + 2 ax + py - b2 - q2 = 0 (C) x2 + y2 + 2 ax + 2 py + b2 + q2 = 0 www.myengg.com The Engineering Universe 5 Entrance Exams ,Engineering colleges in india, Placement details of IITs and NITs ww w. my en gg .co m 6 QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439 (D) None of these 53. The equation of pair of tangents to the circle, x2 + y2 - 2x + 4y + 3 = 0 from (6, - 5), is : (A) 7x2 + 23y2 + 30xy + 66x + 50y − 73 = 0 (B) 7x2 + 23y2 + 30xy − 66x − 50y − 73 = 0 (C) 7x2 + 23y2 − 30xy − 66x − 50y + 73 = 0 (D) None of these 54. If a circle passes through the points of intersection of the co-ordinate axes with the lines, λ x - y + 1 = 0 and x - 2y + 3 = 0, then the value of λ is : (A) 1 (B) 2 (C) 3 (D) 4 55. x = 7, touches the circle x2 + y2 − 4x − 6y − 12 = 0, then the co-ordinates of the point of contact are : (A) (7, 3) (B) (7, 4) (C) (7, 8) (D) (7, 2) 56. The locus of the centre of a circle which touches externally the circle x2 + y2 - 6x - 6y + 14 = 0 and also touches the y-axis, is given by the equation : (A) x2 - 6x - 10y + 14 = 0 (B) x2 - 10x - 6y + 14 = 0 (C) y2 - 6x - 10y + 14 = 0 (D) y2 - 10x - 6y + 14 = 0 57. The equation of the circle which touches the circle, x2 + y2 − 6x + 5y + 17 = 0 externally & to which the lines, x2 − 3xy − 3x + 9y = 0 are normals, is : (A) x2 + y2 - 6x - 2y - 1 = 0 (B) x2 + y2 + 6x + 2y + 1 = 0 (C) x2 + y2 - 6x - 6y + 1 = 0 (D) x2 + y2 - 6x - 2y + 1 = 0 58. The locus of mid-point of the chords of the circle, x2 + y2 − 2x − 2y − 2 = 0 which makes an angle of 120º at the centre is : (A) x2 + y2 - 2x - 2y + 1 = 0 (B) x2 + y2 + x + y - 1 = 0 (C) x2 + y2 - 2x - 2y - 1 = 0 (D) None of these 59. The two circles, x2 + y2 − 2x − 2 = 0 and x2 + y2 − 4x − 6y − 8 = 0 are such that (A) They touch each other (B) They intersect each other (C) One lies inside the other (D) None of these 60. One of the limit point of the co-axial system of circles containing, x2 + y2 − 6x − 6y + 4 = 0 and x2 + y2 - 2x - 4y + 3 = 0 is : (A) (- 1, 1) (B) (- 1, 2) (C) (- 2, 1) (D) (- 2, 2) 61. The centre of the circle, x = − 1 + 2 cos θ, y = 3 + 2 sin θ, is : (A) (1, - 3) (B) (- 1, 3) (C) (1, 3) (D) None of these 62. Tangents drawn from origin to the circle, x2 + y2 − 2 ax − 2 by + b2 = 0 are perpendicular to each other, if : (A) a - b = 1 (B) a + b = 1 (C) a2 = b2 (D) a2 + b2 = 1 63. The line lx + my + n = 0 is normal to the circle, x2 + y2 + 2 gx + 2 fy + c = 0, if (A) lg + mf − n = 0 (B) lg + mf + n = 0 (C) lg − mf − n = 0 (D) lg − mf + n = 0 64. The radical centre of three circles described on the three sides of a triangle as diameter is : (A) The orthocentre (B) The circumcentre www.myengg.com The Engineering Universe 6 Entrance Exams ,Engineering colleges in india, Placement details of IITs and NITs ww w. my en gg .co m 7 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 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 66. The locus of centre of the circle which cuts the circles, x2 + y2 + 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, x2 + y2 + 2 λx + 2 µy + c = 0 touches circle x2 + y2 = r2, if : (A) c = r (λ2 + µ2) (B) r = c (λ2 + µ2) (C) c2 = r2 (λ2 + µ2) (D) r2 = c2 (λ2 + µ2) 68. Line, Ax + By + C = 0 cuts circle, x2 + y2 + ax + by + c = 0 in P & Q and the line A ′x + B ′y + C ′ = 0 cuts the circle x2 + y2 + 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 = a2 and (x + α)2 + (y + β)2 = β2 cuts orthogonally if : (A) a α + b β = b2 + a2 (B) 2 (a α + b β) = b2 + a2 (C) a α + b β = a2 + b2 (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) 2 3 2a (C) a2 6 (D) a2 12 72. The angle between a pair of tangents drawn from a point P to the circle, x2 + y2 + 4x − 6y + 9 sin2 α + 13 cos2 α = 0 is 2 α . The equation of the locus of the point P is : (A) x2 + y2 + 4x - 6y + 4 = 0 (B) x2 + y2 + 4x - 6y - 9 = 0 (C) x2 + y2 + 4x - 6y - 4 = 0 (D) x2 + y2 + 4x - 6y + 9 = 0 73. The intercept on the line y = x, by the circle, x2 + y2 − 2x = 0 is AB . www.myengg.com The Engineering Universe 7 Entrance Exams ,Engineering colleges in india, Placement details of IITs and NITs ww w. my en gg .co m