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Circles

1. If the length of the tangent from any point on the circle (x − 3)2 + (y + 2)2 = 5r2 to the circle (x − 3)2 + (y + 2)2 = r2 is 16 units, then the area between the two circles is
Ans: 256 π

 

2. If the length of transverse common tangent of the circle x2 + y2 = 4 and (x − h)2 + y2 = 1 is 3, then h =
Ans: 3

 

3. An angle subtended by the common chord of x2 + y2 − 4x − 4y = 0 and x2 + y2 = 16 at the origin is
Ans: 

 

4. Minimum radius of circle which is orthogonal to both the circles x2 + y2 − 12x + 35 = 0 and x2 + y2 + 4x + 3 = 0 is
Ans: 

 

5. The range of values of 'a' such that the angle θ between the pair of tangents drawn from (a, 0) to the circle x2 + y2 = 1 satisfies   <  < π is
Ans: (−, −1)  (1, )

6. If r1 and r2 are radii of the circles whose centres are at a distance '1' unit from the origin and touches the lines y = 0 and y =  (x + 1), then    = 
Ans: 1

 

7. If 3x + 4y + 15 = 0 cuts equal intercepts between x2 + y2 = 13, x2 + y2 = r2 and x2 + y2 = 45 ( < r < ), then r =
Ans: 5

 

8. The locus of all points P whose farthest and shortest distances to the circle (x − a)2 + (y + b)2 = (a + b)2 are 2a, 2b (a > b > 0) is
Ans: (x − a)2 + (y − b)2 = a2 + b2

 

9. Let S1 ≡ x2 + y2 − 4x − 8y + 4 = 0 and S2 be its image in the line y = x. The equation of the circle touching y = x at (1, 1) and orthogonal to S2 is
Ans: x2 + y2 + x − 5y + 2 = 0

 

10. x2 + y2 + 6x + 8y = 0 and x2 + y2 − 4x − 6y − 12 = 0 are the equations of the two circles. Equation of one of their common tangent is
Ans: 7x − 5y + 1 − 5 = 0

 

11. The equation of chord of the circle x2 + y2 − 6x − 4y − 12 = 0 which passes through the origin such that the origin divides it in the ratio 3 : 2 is
Ans: x + y = 0; 7y + 17x = 0

 

12. The points 'A', 'B' are the feet of O (0, 0) on x − 2y + 1 = 0, 2x − y − 1 = 0 respectively, then the circum radius of the ∆OAB is
Ans:  

 

13. The locus of the image of the point (2, 3) with respect to the line (x − 2y + 3) + λ (2x − 3y + 4) = 0 (λ ϵ R) is
Ans: x2 + y2 − 2x − 4y + 3 = 0

 

14. Let A (1, 2), B (3, 4) be two points and C(x, y) be a point such that area of ∆ABC is 3 sq.units and (x − 1)(x − 3) + (y − 2)(y − 4) = 0, then maximum number of positions of 'C' in the XY plane is
 A) 2     B) 4    C) 8     D) No such 'C' exists
Ans: No such 'C' exist

 

15. If the circle x2 + y2 + 4x + 22y + c = 0 bisects the circumference of the circle x2 + y2 − 2x + 8y − d = 0, then c + d =
Ans: 50

 

16. The equation of a circle through the intersection of x2 + y2 + 2x = 0 and x − y = 0 having minimum radius is
Ans: x2 + y2 − x − y = 1

 

17. From a point 'P' outside a circle with centre at 'C' tangents PA and PB are drawn such that   , then length of chord AB =
Ans: 8 units

 

18. The chord through (2, 1) to the circle x2 + y2 − 2x − 2y + 1 = 0 is bisected at the point , then the value of 'α' =
    A)     B) 1     C) 0      D) None of the above
Ans: None of the above

 

19. The curve x2 − y − x + 1 = 0 intersects X − axis at 'A' and 'B'. A circle is drawn passing through A and B. The length of tangent drawn from the origin to this circle is
     A)     B)     C)     D) None of these
Ans: None of these

 

20. The locus of the mid points of chords of the circle x2 + y2 = 9 such that segment intercepted by the chord on the curve y2 − 4x − 4y = 0 subtends a right angle at the origin is
Ans: x2 + y2 − 4x − 4y = 0


21. Tangents are drawn to the circle x2 + y2 = 10 at the points where it is met by the circle x2 + y2 − 6x − 4y + 10 = 0, then the point of intersection of these tangents is
Ans: (3, 2)

 

22. The equation of the circle described on the common chord of the circles x2 + y2 − 4x − 5 = 0 and x2 + y2 + 8y + 7 = 0 as a diameter is
Ans: x2 + y2 − 2x + 4y + 1 = 0

 

23. The area bounded by the circles x+ y2 = r2; r = 1, 2 and the rays given by 2x2 − 3xy − 2y2 = 0; y > 0 is
Ans: 

 

24. The points A, B are the feet of O(0, 0) on x − 2y + 1 = 0; 2x − y − 1 = 0 respectively, then the circum radius of ∆OAB is
Ans: 

 

25. If a circle of radius '3' pass through origin 'O' and meets co-ordinate axes at A and B, then the locus of the centroid of triangle OAB is
Ans: x2 + y2 = 4

 

26. Tangents PA and PB are drawn to the circle (x + 2)2 + (y − 2)2 = 1 from the points which lie on the line y = x, then the locus of circumcentre of ∆PAB is
Ans: y = x + 2

 

27. There are exactly two chords of the circle x2 + y2 = 100 that passes through (1, 7) and subtends an angle  at the origin, then the angle between these two chords is
Ans: 

 

28. 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)    B) (1 − , − 2)    C) (1, − 2 + )    D) None of these
Ans: None of these

 

29. The coordinates of a point 'P' on the circle x2 + y2 − 4x − 6y + 9 = 0 such that  is minimum where 'O' is the origin and OX is the X - axis are
Ans: 

 

30. If the circles (x − 1)2 + (y − 3)2 = r2 and x2 + y2 − 8x + 2y + 8 = 0 intersect in two distinct points, then
Ans: 2 < r < 8

 

31. If one of the diameters of the circle x2 + y− 2x − 6y + 6 = 0 is a chord to the circle with centre at (2, 1), then the radius of the circle is
Ans: 2

 

32. AB is a chord of the circle x2 + y2 = 9. The tangents at A and B intersect at C. If (1, 2) is the mid point of AB, then the area of triangle ABC is
Ans:    sq.units

 

33. Tangents are drawn to the circle x2 + y2 = 1 at the points where it is met by the circles x2 + y2 − (λ + 6)x + (8 − 2λ)y − 3 = 0, λ ϵ R then locus of the point of intersection of these tangents is
Ans: 2x − y + 10 = 0

 

34. Let PQ and RS be tangents at the extremities of a diameter PR of a circle of radius 'r' such that PS and RQ intersect at a point 'X' on the circumference of the circle, then 2r equals to
Ans: 

 

35. P(1, 1) is a fixed point on the circle x2 + y2 − 6x − 4y + 8 = 0 and A, B are moving on the circumference of the same circle such that PA = PB = d ( > 0), the equation of the secant line  when 'd' is maximum is
Ans: 2x + y − 13 = 0

 

36. A variable circle C has the equation x2 + y2 − 2(t2 − 3t + 1)x − 2(t2 + 2t)y + t = 0 where t is a parameter, then locus of the centre of the circle is
Ans: a parabola

 

37. If x2 + y2 = 16; x2 + y2 = 36 are two circles and P and Q moves respectively on these circles such that PQ = 4, then the locus of mid point of PQ is a circle of radius
Ans: 

 

38. An isosceles triangle with base '24' and legs 15 each, is inscribed in a circle with centre at (−1, 1), then locus of the centroid of that triangle is
Ans: 4(x2 + y2) + 8x − 8y − 161 = 0

 

39. The normal of the circles (x − 2)2 + (y − 1)2 = 1 which bisect the chord cut off by the line x − 2y + 3 = 0 is
Ans: 2x + y + 3 = 0

 

40. The circle x2 + y+ 4x + 8y + 5 = 0 intersects the line 3x − 4y = m at two distinct points if
Ans: −35 < m < 15

 

Multiple Correct Answer Type Questions 

41. If the circle C1: x2 + y2 = 16 intersects another circle C2 of radius '5' in such a manner that the common chord is of maximum length and has a slope equal to , then the coordinates of the centre of C2 are
   A)     B)      C)        D) 
Ans: A, B

 

42. If the circle x2 + y2 + 2gx + 2fy + c = 0 cuts each of the circles x2 + y2 = 4; x2 + y2 − 6x − 8y + 10 = 0 and x+ y2 + 2x − 4y − 2 = 0 at the extremities of a diameter, then
   A) c = − 4      B) g + f = c − 1    C) g2 + f2 − c = 17       D) gf = 6
Ans: A, B, C, D

 

43. Tangents are drawn to the circle x2 + y2 = 50 from a point 'P' lying on the X - axis. These tangents meet the Y - axis at points 'P1' and 'P2'. Possible coordinates of P 'so' that area of triangle PP1P2 is minimum are
     A) (10, 0)     B) (10, 0)     C) (−10, 0)    D) (−10, 0)
Ans: A, C

 

44. The locus of the point of intersection of the tangents at the extremities of a chord of the circle x2 + y2 = b2 which touches the circle x2 + y2 − 2by = 0 passes through the point
      A)      B) (0, b)     C) (b, 0)     D) 
Ans: A, C

 

45. C1, C2 are two circles of radii a, b (a < b) touching both the coordinate axes and have their centres in the I quadrant, then the true statements are 
     A) if C1, C2 touch each other, then   = 3 + 2
     B) if C1, C2 are orthogonal, then  = 2 + 
    C) if C1, C2 intersect in such a way that their common chord has maximum length, then   = 3 
    D) if C2 passes through centre of C1, then  = 2 + 
Ans: A, B, C, D

 

Passage - I (46 - 48)
 A variable circle of radius 2 units rolls outside the circle x2 + y2 + 4x = 0 if 'C' and 'C1' denotes the centre of respective circles, then

 

46. Locus of centre of variable circle i.e., locus of 'C' is
Ans: x2 + y2 + 4x − 12 = 0

 

47. If the line joining C and C1 makes an angle of 60° with X − axis, then equation of common tangents of the circle can be
Ans:  x − y ±   = 0;  y + x − 2 = 0

 

48. The area of region formed by the common tangents and the line x +  y + 2 = 0 is ....... sq.units
Ans: 4

 

Passage (49 − 51)
P(a, 5a) and Q(4a, a) are two points. Two circles are drawn through these points touching the axis of  Y

 

49. Centre of these circles are at
Ans: 

 

50. Angle of intersection of these circles is-
Ans: 

 

51. If C1, C2 are centres of these circles, then area of ∆OC1C2 where 'O' is origin is
Ans: 5a2

 

MATRIX - MATCH TYPE 

52. Consider the equation of the circle x2 + y2 = 9 and a line y = 1 which divides the circle into two regions, let small region be R1 and other be R2. Then match the following.


 

Ans: (i) S (ii) R (iii) P (iv) Q

 

53. A (−2, 0), B (2, 0) are two fixed points and P is a point such that PA − PB = 2. Let S be the circle x2 + y2 = r2, then match the following.

Ans: (i) P (ii) S (iii) R (iv) Q

Posted Date : 17-02-2021

గమనిక : ప్రతిభ.ఈనాడు.నెట్లో వచ్చే ప్రకటనలు అనేక దేశాల నుండి, వ్యాపారస్తులు లేదా వ్యక్తుల నుండి వివిధ పద్ధతులలో సేకరించబడతాయి. ఆయా ప్రకటనకర్తల ఉత్పత్తులు లేదా సేవల గురించి ఈనాడు యాజమాన్యానికీ, ఉద్యోగస్తులకూ ఎటువంటి అవగాహనా ఉండదు. కొన్ని ప్రకటనలు పాఠకుల అభిరుచిననుసరించి కృత్రిమ మేధస్సు సాంకేతికతతో పంపబడతాయి. ఏ ప్రకటనని అయినా పాఠకులు తగినంత జాగ్రత్త వహించి, ఉత్పత్తులు లేదా సేవల గురించి తగిన విచారణ చేసి, తగిన జాగ్రత్తలు తీసుకొని కొనుగోలు చేయాలి. ఉత్పత్తులు / సేవలపై ఈనాడు యాజమాన్యానికి ఎటువంటి నియంత్రణ ఉండదు. కనుక ఉత్పత్తులు లేదా సేవల నాణ్యత లేదా లోపాల విషయంలో ఈనాడు యాజమాన్యం ఎటువంటి బాధ్యత వహించదు. ఈ విషయంలో ఎటువంటి ఉత్తర ప్రత్యుత్తరాలకీ తావు లేదు. ఫిర్యాదులు తీసుకోబడవు.

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