# 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

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

Circles and System of Circles

1. The sum of the square of length of the chord intercepted by the line x + y = n (n∈N) on the circle x2 + y2 = 4 is
1) 11         2) 22            3) 33          4) 44

2. Let C be the circle with centre at (1 1) and radius 1. If T is another circle centre at (0, α) passing through (0, 0),  touching the circle C externally then the radius of the circle T is
1) 1/      2) 1/2             3) 1/3               4) 1/4

3. If a line passes through the point P(1 -2) and cuts the circle x2 + y2 - x - y = 0 at A, B then the maximum value of PA + PB is
1) √26       2) √17         3) 3         4) 2 √3

4. Let x2 + y2 - ax - y = 0 be a circle. If the two chords from (a 1) bisected by x - axis are drawn to the circle then the condition is
1) a2 > 8          2) a2< 8           3) a2 > 4            4) a2< 4

5. The radius of the least circle passing through the point P (8, 4) and cutting the circle x2 + y2 - 40 = 0 orthogonally is
1) √3        2)  √5           3) √7          4) 3

6. If 3, 4 are the radii and 5 is the distance between the centres of two intersecting circles then the length of common chord of the circle is
1) 12/5          2) 24/25          3) 24/5            4) 5/24

7. If the circles x2 + y2 = 2 and x2 + y2 - 4x - 4y + λ = 0 have exactly three real common tangents then λ =
1) -10          2) 6           3) -6        4) 10

8.

distinct points on a circle then m1 m2 m3 m4 =
1) 4          2) -1            3) 1           4) -4

9. If the distance between the centres of two circles of radii 3, 4 is 25. Then the length of the transverse common tangent is
1) 24          2) 17            3) 9         4) 26

Numerical Value Type

10. ABCD is a square of unit area. A circle is tangent to two sides of square ABCD and passes through exactly one of its vertices. The radius of the circle is

11. If the circle 3x2 + 3y2 + 10x + y - 27 = 0 bisects the circumference of the circle x2 + y2 = r then the value of  (r/3)2

12. A right angled isosceles triangle is inscribed in the circle x2 + y2 - 4x - 2y - 4 = 0, then length of its side is

13. The perpendicular distance from (1, -2) from the common chord of two circles x2 + y2 - 5x + 4y - 2 = 0 and x2 + y2 - 2x + 8y + 3 = 0 is

14. A circle is inscribed in an equilateral triangle of side ‘6’. Then the area of any square inscribed in this circle is

15. The shortest distance from (-2  14) to the circle x2 + y2 - 6x - 4y -12 = 0 is

16. If the lines 3x - 4y + 4 = 0, 6x - 8y - 7 = 0 are tangent to a circle, then the radius of the circle is

17. If an equilateral triangle is inscribed in the circle x2 + y2 - 6x - 4y + 5 = 0 and the length of side the triangle is K√6 then K =

Key: 1-2; 2-4; 3-1; 4-1; 5-2; 6-3; 7-3; 8-3; 9-1; 10-0.59; 11-9; 12-4.23; 13-0; 14-6; 15-8; 16-0.75; 17-2.

Posted Date : 18-03-2021

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గమనిక : ప్రతిభ.ఈనాడు.నెట్‌లో కనిపించే వ్యాపార ప్రకటనలు వివిధ దేశాల్లోని వ్యాపారులు, సంస్థల నుంచి వస్తాయి. మరి కొన్ని ప్రకటనలు పాఠకుల అభిరుచి మేరకు కృత్రిమ మేధస్సు సాంకేతికత సాయంతో ప్రదర్శితమవుతుంటాయి. ఆ ప్రకటనల్లోని ఉత్పత్తులను లేదా సేవలను పాఠకులు స్వయంగా విచారించుకొని, జాగ్రత్తగా పరిశీలించి కొనుక్కోవాలి లేదా వినియోగించుకోవాలి. వాటి నాణ్యత లేదా లోపాలతో ఈనాడు యాజమాన్యానికి ఎలాంటి సంబంధం లేదు. ఈ విషయంలో ఉత్తర ప్రత్యుత్తరాలకు, ఈ-మెయిల్స్ కి, ఇంకా ఇతర రూపాల్లో సమాచార మార్పిడికి తావు లేదు. ఫిర్యాదులు స్వీకరించడం కుదరదు. పాఠకులు గమనించి, సహకరించాలని మనవి.