1. The equation of a circle with centre (5, 4) and touch the y-axis is

1. 1. \(x^{2} + y^{2} – 10x – 8y – 16 = 0\)
2. 2. \(x^{2} + y^{2} – 10x – 8y – 61 = 0\)
3. 3. \(x^{2} + y^{2} + 10x + 8y + 16 = 0\)
4. 4. \(x^{2} + y^{2} – 10x – 8y + 16 = 0\)

2. If \(x^{2} + y^{2} + 6x + 2ky + 25 = 0\) to touch y-axis, then \(\mathbf{k} =\)

1. 1. \(\pm 20\)
2. 2. \(-1, -5\)
3. 3. \(\pm 5\)
4. 4. 4

3. The point on the circle \(x^{2} + y^{2} = 4\) whose distance from the line \(4x + 3y – 12 = 0\) is \(4 / 5\) units is equal to

1. 1. \(\left(\frac{12}{25}, \frac{36}{25}\right)\)
2. 2. \((4, 0)\)
3. 3. \((2, 0)\)
4. 4. \(\left(\frac{-14}{25}, \frac{48}{25}\right)\)

4. The equation of the circle passing through (0,0) and which makes intercepts a and b on the co-ordinate axes is

1. 1. \(x^{2} + y^{2} + ax + by = 0\)
2. 2. \(x^{2} + y^{2} + ax – by = 0\)
3. 3. \(x^{2} + y^{2} – ax + by = 0\)
4. 4. \(x^{2} + y^{2} – ax – by = 0\)

5. The value of m+n if the circumference of the circle \(x^{2} + y^{2} + 8x + 8y – m = 0\) is bisected by the circle \(x^{2} + y^{2} – 2x + 4y + n = 0\)

1. 1. -56
2. 2. 56
3. 3. 50
4. 4. -34

6. The equation of normal at (1, 1) to the circle \(x^{2} + y^{2} – x – 3y – 4 = 0\) is

1. 1. \(x + y – 2 = 0\)
2. 2. \(2x – y – 1 = 0\)
3. 3. \(x – y + 2 = 0\)
4. 4. \(x – y – 2 = 0\)

7. The length of the diameter of the circle \(x^{2} + y^{2} – 6x – 8y = 0\) is units

1. 1. 5
2. 2. 10
3. 3. 15
4. 4. 20

8. If \(3x + y + k = 0\) is a tangent to the circle \(x^{2} + y^{2} = 10\) then \(\mathbf{k} =\)

1. 1. \(\pm 7\)
2. 2. \(\pm 5\)
3. 3. \(\pm 9\)
4. 4. \(\pm 10\)

9. The equations of tangents to the circle from the point (4, -2) are

1. 1. \(x + y = 2\), \(3x + 2y = 16\)
2. 2. \(5x + y = 18\), \(3x – y = 4\)
3. 3. \(3x + y = 10\), \(x – 3y = 10\)
4. 4. \(5x – y = 4\), \(x + y = 0\)

10. If the two circles \((x – 1)^{2} + (y – 3)^{2} = r^{2}\) and \(x^{2} + y^{2} – 8x + 2y + 8 = 0\) intersect in two different points, then what can we conclude about \(r?\)

1. 1. \(r < 2\)
2. 2. \(r = 2\)
3. 3. \(r > 2\)
4. 4. \(2 < r < 8\)

11. The polar of (1, -2) with respect to \(x^{2} + y^{2} – 10x – 10y + 25 = 0\)

1. 1. \(4x + 7y + 30 = 0\)
2. 2. \(4x + 7y – 30 = 0\)
3. 3. \(4x – 7y + 30 = 0\)
4. 4. \(x + y = 0\)

12. The equation of the smallest circle passing through the intersection of the line \(x + y = 1\) and the circle \(x^{2} + y^{2} = 9\) is

1. 1. \(x^{2} + y^{2} – 9 – (x + y + 1) = 0\)
2. 2. \(x^{2} + y^{2} – 9 – (x + y – 1) = 0\)
3. 3. \(x^{2} + y^{2} – 9 – x + y – 1 = 0\)
4. 4. \(x^{2} + y^{2} – 9 + x + y – 1 = 0\)

13. A circle is drawn touching the x-axis, with its centre at the point of reflection of (m,n) on the line y- \(\mathbf{x} = 0\) . Then the equation of the circle is

1. 1. \(x^{2} + y^{2} – 2m x – 2n y + m^{2} = 0\)
2. 2. \(x^{2} + y^{2} – 2m x + 2n y + m^{2} = 0\)
3. 3. \(x^{2} + y^{2} + 2n x – 2m y – n^{2} = 0\)
4. 4. \(x^{2} + y^{2} – 2n x – 2m y + n^{2} = 0\)

14. To which point the origin is to be shifted in order to eliminate first powers of \(x\) and \(y\) \((x^{\prime}and y^{\prime}terms)\) from the equation \(4x^{2} + 9y^{2} – 8x + 36y + 4 = 0?\)

1. 1. (1,2)
2. 2. (-1,2)
3. 3. (1,-2)
4. 4. (-1,-3)

15. 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

1. 1. \(\sqrt{PQ \cdot RS}\)
2. 2. \(\frac{PQ + RS}{2}\)
3. 3. \(\frac{2PQ \cdot RS}{PQ + RS}\)
4. 4. \(\sqrt{\frac{(PQ)^2 + (RS)^2}{2}}\)

16. The area of the equilateral triangle inscribed in the circle \(x^{2} + y^{2} + 6x + 2y – 28 = 0\) is square units

1. 1. \(\frac{27\sqrt{3}}{2}\)
2. 2. \(\frac{37\sqrt{3}}{2}\)
3. 3. \(\frac{31\sqrt{3}}{2}\)
4. 4. \(\frac{57\sqrt{3}}{2}\)

17. The equation of the circle with centre (2,3) and touching the line \(3x – 4y + 1 = 0\) is

1. 1. \(x^{2} + y^{2} + 4x + 4y + 12 = 0\)
2. 2. \(x^{2} + y^{2} – 4x – 6y – 14 = 0\)
3. 3. \(x^{2} + y^{2} – 4x – 6y + 14 = 0\)
4. 4. \(x^{2} + y^{2} – 4x – 6y + 12 = 0\)

18. The length of the chord intercepted by the circle \(x^{2} + y^{2} – 6x + 8y – 5 = 0\) on the line \(2x – y = 5\) is equal to- units

1. 1. 10
2. 2. 12
3. 3. 7
4. 4. 8

19. The circle \(x^{2} + y^{2} – 6x – 10y + p = 0\) neither intersects nor touch the coordinate axes and the point (1,4) lies inside the circle. Then the range of possible values of \(\mathbf{\bar{\rho}}^{*}\mathbf{\bar{\rho}}\) is

1. 1. \(23< \mathrm{p}< 25\)
2. 2. \(25< \mathrm{p}< 29\)
3. 3. \(21< \mathrm{p}< 23\)
4. 4. \(12< \mathrm{p}< 21\)

20. A circle passes through the centre of another circle \(x^{2} + y^{2} – 3x – 4y – 1 = 0\) and whose centre is (5,2). Then the equation of this circle is

1. 1. \(4x^{2} + 4y^{2} – 40x – 16y + 67 = 0\)
2. 2. \(3x^{2} + 3y^{2} – 40x – 16y + 67 = 0\)
3. 3. \(2x^{2} + 2y^{2} – 40x – 16y + 67 = 0\)
4. 4. \(x^{2} + y^{2} – 10x – 4y + 67 = 0\)

21. If the chord of contact of tangents from a point A to a given circle passes through B, then the circle with AB as a diameter will

1. 1. Touch the given circle internally
2. 2. Cut the given circle orthogonally
3. 3. Touch the given circle externally
4. 4. Neither intersect nor touch the given circle

22. Find the equation of circle having normals (x-1) (y-2)=0 and tangent \(3x + 4y = 6?\)

1. 1. \((x – 1)^{2} + (y – 2)^{2} = 1\)
2. 2. \((x – 2)^{2} + (y – 1)^{2} = 1\)
3. 3. \((x + 1)^{2} + (y + 2)^{2} = 1\)
4. 4. \((x + 2)^{2} + (y + 1)^{2} = 1\)

23. If \(y = \sqrt{3} x + k_{1}\) and \(y = \sqrt{3} x + k_{2}\) are two parallel tangents of a circle of radius 2 units, then \(|k_{1} – k_{2}|\) is equal to

1. 1. 1
2. 2. 8
3. 3. 4
4. 4. 2

24. The centre of a circle is \((2, – 3)\) and the circumference is \(10\pi\) . Then its equation is

1. 1. \(x^{2} + y^{2} + 4x + 6y + 12 = 0\)
2. 2. \(x^{2} + y^{2} – 4x + 6y + 12 = 0\)
3. 3. \(x^{2} + y^{2} – 4x + 6y – 12 = 0\)
4. 4. \(x^{2} + y^{2} – 4x – 6y – 12 = 0\)

25. Find the maximum distance of the point K(10,7) from the circle \(x^{2} + y^{2} – 4x – 2y – 20 = 0\)

1. 1. 25
2. 2. 10
3. 3. 15
4. 4. 5

26. In \(\Delta ABC\) \(\angle A = 90^{\circ}\) and co-ordinates of points B and C are \((2, – 4)\) and \((1,5)\) . Then the equation of the circumcircle of \(\Delta ABC\) is

1. 1. \(x^{2} + y^{2} + 3x + y + 18 = 0\)
2. 2. \(x^{2} + y^{2} – 3x + y – 18 = 0\)
3. 3. \(x^{2} + y^{2} – 3x – y – 18 = 0\)
4. 4. \(x^{2} + y^{2} + 3x – y + 18 = 0\)

27. The equation of the circle circumscribing the triangle formed by the straight lines \(x + y = 6,2x + y = 4\) and \(x + 2y = 5\) is given by

1. 1. \(x^{2} + y^{2} + 17x + 19y + 50 = 0\)
2. 2. \(x^{2} + y^{2} – 17x – 19y + 50 = 0\)
3. 3. \(x^{2} + y^{2} + 17x – 19y – 50 = 0\)
4. 4. \(x^{2} + y^{2} – 17x + 19y – 50 = 0\)

28. If the point \((1,4)\) lies inside the circle \(x^{2} + y^{2} – 6x – 10y + p = 0\) and the circle does not touch or intersect the coordinates axes, then

1. 1. \(0< \mathrm{p}< 34\)
2. 2. \(25< \mathrm{p}< 29\)
3. 3. \(9< \mathrm{p}< 25\)
4. 4. \(7< \mathrm{p}< 29\)

29. In \(\Delta\) ABC, the circle that touches the sides BC internally and other two sides AB and AC externally, is called….

1. 1. Ex circle opposite to angle A
2. 2. Inscribed circle opposite to angle A
3. 3. Circumcircle of the triangle
4. 4. No such circle exists

30. The length of the tangent drawn from the midpoint of the line joining the origin and the point \((4, – 4)\) , to the circle \(2x^{2} + 2y^{2} – y = 0\) is – units

1. 1. \(3\sqrt{2}\)
2. 2. \(\sqrt{2}\)
3. 3. \(\sqrt{10}\)
4. 4. 3

31. For the circle \(x^{2} + y^{2} – 9 = 0\) , find the equation of the chord having \((1,2)\) as its mid- point.

1. 1. \(x + 2y + 5 = 0\)
2. 2. \(x – 3y – 5 = 0\)
3. 3. \(x – 3y + 9 = 0\)
4. 4. \(x + 2y – 9 = 0\)

32. The area of the quadrilateral formed by the tangents from the point \((4,5)\) to the circle \(x^{2} + y^{2} – 4x – 2y – 11 = 0\) , with a pair of radii joining the points of contact of these tangents is

1. 1. 4
2. 2. 6
3. 3. 8
4. 4. 10

33. A square is inscribed in the circle \(x^{2} + y^{2} – 2x + 4y – 93 = 0\) with its sides parallel to the co-ordinate axes. Then which among the following can be one of the vertices of the square?

1. 1. (5,8)
2. 2. (8,5)
3. 3. (8,-5)
4. 4. (-8,5)

34. Find the equation of a circle with radius 5 units and touching the circle \(x^{2} + y^{2} – 2x – 4y – 20 = 0\) at the point \((5,5)\) .

1. 1. \(x^{2} + y^{2} – 18x – 16y + 120 = 0\)
2. 2. \(x^{2} + y^{2} + 18x + 16y – 120 = 0\)
3. 3. \(x^{2} + y^{2} – 18x + 16y – 120 = 0\)
4. 4. \(x^{2} + y^{2} + 18x + 16y + 120 = 0\)

35. The length of the tangent from (6, 8) to the circle \(x^{2} + y^{2} = 4\) is

1. 1. \(\sqrt{6}\)
2. 2. \(2\sqrt{6}\)
3. 3. \(4\sqrt{6}\)
4. 4. \(5\sqrt{6}\)

36. If the length of the tangent from \((f,g)\) to the circle \(x^{2} + y^{2} = 6\) be twice the length of the tangent from the same point to the circle \(x^{2} + y^{2} + 3x + 3y = 0\) , then \(f^{2} + g^{2} + 4f + 4g + 2\) is equal to

1. 1. -1
2. 2. 1
3. 3. 0
4. 4. -2

37. The radius of any circle touching the lines \(3x – 4y + 5 = 0\) , \(6x – 8y – 9 = 0\)

1. 1. 1
2. 2. 23/15
3. 3. 20/19
4. 4. 19/20

38. The equation of pair of straight lines parallel to x-axis and touching the circle \(x^{2} + y^{2} – 6x – 4y – 12 = 0\) is

1. 1. \(y^{2} – 4y – 21 = 0\)
2. 2. \(y^{2} + 4y – 21 = 0\)
3. 3. \(y^{2} – 4y + 21 = 0\)
4. 4. \(y^{2} + 4y + 21 = 0\)

39. Find the area of the circle \((x + 1)(x + 2) + (y – 1)(y + 3) = 0\)

1. 1. \(\frac{17\pi}{4}\)
2. 2. \(\frac{17\pi}{2}\)
3. 3. \(\frac{2\pi}{17}\)
4. 4. \(\frac{\pi}{3}\)

40. The equation of a circle which touches the x-axis and whose centre is (1, 2) is

1. 1. \((x – 2)^{2} + (y – 1)^{2} = 4\)
2. 2. \((x – 1)^{2} + (y – 2)^{2} = 4\)
3. 3. \((x – 1)^{2} + (y + 2)^{2} = 4\)
4. 4. \((x + 2)^{2} + (y – 1)^{2} = 4\)

41. If a line drawn from a fixed point \(M(a,b)\) cuts the circle \(x^{2} + y^{2} = k^{2}\) at C and D, then \(MC \times MD\) is equal to

1. 1. \(a^{2} + b^{2} + k^{2}\)
2. 2. \(a^{2} + b^{2} – k^{2}\)
3. 3. \(a^{2} – b^{2} – k^{2}\)
4. 4. \(k^{2}\)

42. The radius of the circle \(2x^{2} + 2y^{2} – 3x + 2y – 1 = 0\) is units.

1. 1. \(\frac{\sqrt{21}}{2}\)
2. 2. \(\frac{\sqrt{21}}{4}\)
3. 3. \(\frac{21}{4}\)
4. 4. \(\frac{\sqrt{5}}{4}\)

43. If one end of the diameter of \(x^{2} + y^{2} – 2x – 6y – 15 = 0\) is (4,1) then the co-ordinates of the other end is

1. 1. (5,-2)
2. 2. (-2,5)
3. 3. (1,3)
4. 4. (-2,-5)

44. The angle between the pair of tangents drawn from (1,3) to the circle \(x^{2} + y^{2} – 2x + 4y – 11 = 0\) is

1. 1. \(\sin^{-1}\left(\frac{24}{25}\right)\)
2. 2. \(\sin^{-1}\left(\frac{7}{25}\right)\)
3. 3. \(\cos^{-1}\left(\frac{24}{25}\right)\)
4. 4. \(\tan^{-1}\left(\frac{7}{24}\right)\)

45. If (1,a),(b,2) are conjugate points with respect to the circle \(x^{2} + y^{2} = 25\) ,then \(4a + 2b =\)

1. 1. 25
2. 2. 50
3. 3. 75
4. 4. 100

46. A circle cuts off positive intercepts 5 and 6 on the x and y axes respectively, and passes through the origin. Then the equation of the circle is

1. 1. \(x^{2} + y^{2} + 5x + 6y = 0\)
2. 2. \(x^{2} + y^{2} – 5x + 6y = 0\)
3. 3. \(x^{2} + y^{2} – 5x – 6y = 0\)
4. 4. \(x^{2} + y^{2} + 5x – 6y = 0\)

47. The length of the chord intercepted by the circle \(x^{2} + y^{2} – 4x + 4y + 3 = 0\) on the line \(x = 3y + 13\) is units.

1. 1. \(\sqrt{10}\)
2. 2. \(\sqrt{20}\)
3. 3. \(5\sqrt{2}\)
4. 4. \(2\sqrt{5}\)

48. The equation of the tangent to the circle \(x^{2} + y^{2} = 5\) at (-3, 4) is

1. 1. \(4x – 3y + 25 = 0\)
2. 2. \(x + y – 1 = 0\)
3. 3. \(3x + 4y = 0\)
4. 4. \(3x – 4y + 5 = 0\)

49. The equation of a normal to the circle \(x^{2} + y^{2} – 2x = 0\) that is parallel to the line \(x + 2y – 3 = 0\) is

1. 1. \(x – 2y = 1\)
2. 2. \(x + 2y = 1\)
3. 3. \(x + 2y + 1 = 0\)
4. 4. \(x – 2y – 2 = 0\)

50. If the circle \(x^{2} + y^{2} – 6x + 2y = 28\) cuts off a chord of length \(\lambda\) units on the line \(2x – 5y + 18 = 0\) then the value of \(\lambda\) is

1. 1. 3
2. 2. 6
3. 3. 12
4. 4. 9

51. Consider the circles \(S_{1}:x^{2} + y^{2} + 2x + 8y – 23 = 0\) and \(S_{2}:x^{2} + y^{2} – 4x + 10y + 19 = 0\) . If the polar of the centre of a circle with respect to the another circle are \(L_{1}\) and \(L_{2}\) ,then \(L_{1},L_{2}\) are

1. 1. parallel and separated by a distance of \(4\sqrt{10}\) units
2. 2. perpendicular and intersect at (1,3)
3. 3. perpendicular and intersect at (1,-5)
4. 4. parallel and separated by a distance of \(2\sqrt{10}\) units

52. If the circle \(S = x^{2} + y^{2} – 4 = 0\) intersects another circle \(S^{\prime} = 0\) of radius \(\frac{5\sqrt{2}}{2}\) in such a manner that the common chord is of maximum length with slope equal to \(\frac{1}{4}\) , then the centre of \(S^{\prime} = 0\) is

1. 1. \((-1,4)or(1, – 4)\)
2. 2. \(\left(-\frac{\sqrt{2}}{2}, 2\sqrt{2}\right)or\left(\frac{\sqrt{2}}{2}, -2\sqrt{2}\right)\)
3. 3. \(\left(-2\sqrt{2},\frac{\sqrt{2}}{2}\right)or\left(2\sqrt{2},\frac{-\sqrt{2}}{2}\right)\)
4. 4. \((4, -1)or(-4,1)\)

53. If A(-1,3) and B(5,3) are points on a circle C and the chord AB subtends an angle \(\frac{\pi}{4}\) at a point P on C, then the equation of such a circle C is

1. 1. \(x^{2} + y^{2} – 4x + 12y + 22 = 0\)
2. 2. \(x^{2} + y^{2} – 4x – 12y + 22 = 0\)
3. 3. \(x^{2} + y^{2} – 4x – 12y – 22 = 0\)
4. 4. \(3x^{2} + 3y^{2} – 12x – 36y – 66 = 0\)

54. The midpoint of the chord of the circle \(x^{2} + y^{2} – 6x + 4y – 12 = 0\) drawn parallel to the tangent at (-1,1) and at a distance of one unit from the tangent is

55. If the angle between the pair of tangents drawn to the circle \(x^{2} + y^{2} – 2x + 4y + 3 = 0\) from (6,-5) is \(\theta\) then \(\tan \theta =\)

1. 1. \(\frac{5}{8}\)
2. 2. \(\frac{15}{8}\)
3. 3. \(\frac{8}{15}\)
4. 4. \(\frac{19}{8}\)

56. Two points from the set of concyclic points of the circle passing through (1,1),(2,- 1),(3,2) is

1. 1. \(\left(\frac{5}{2} +\sqrt{\frac{5}{2}},\frac{1}{2} +\sqrt{\frac{5}{2}}\right),\left(\frac{5}{2},\frac{1}{2} +\sqrt{\frac{5}{2}}\right)\)
2. 2. \(\left(\frac{5}{2} +\sqrt{\frac{5}{2}},\frac{1}{2}\right),\left(\frac{5 + \sqrt{5}}{2},\frac{1 + \sqrt{5}}{2}\right)\)
3. 3. \(\left(\frac{5 + \sqrt{5}}{2},\frac{1 + \sqrt{5}}{2}\right),\left(\frac{5}{2} +\sqrt{\frac{5}{2}},\frac{1 + \sqrt{5}}{4}\right)\)

64. If a circle of radius r touches the positive coordinate axes and also the circle \(x^{2} + y^{2} – 12x – 10y + 52 = 0\) externally, then the distance between the centres of the two circles is

1. 1. 7
2. 2. 5
3. 3. 6
4. 4. 8

65. If the circles \(x^{2} + y^{2} – 2x + k = 0\) and \(x^{2} + y^{2} + 4x + 6y + 4 = 0\) touch each other externally, then the point of contact of two circles is

1. 1. \(\left(-\frac{1}{5}, -\frac{3}{5}\right)\)
2. 2. \(\left(-\frac{1}{3}, -\frac{1}{3}\right)\)
3. 3. \((-1, -3)\)
4. 4. \((-1, -1)\)

66. If PA and PB are the tangents drawn from the point \(\mathbf{p}(1,1)\) to the circle \(x^{2} + y^{2} + gx + gy – 2 = 0\) with C as the centre, then the area (in sq. units) of the quadrilateral PACB is

1. 1. \(2\sqrt{g}\)
2. 2. \(\sqrt{g^{3} – 4g}\)
3. 3. \(\sqrt{g^{3} + 4g}\)
4. 4. \(\sqrt{\frac{g^{3}}{2} + 4g}\)

67. The point(s) of intersection of the common tangents of the two circles \(x^{2} + y^{2} – 8x – 6y + 21 = 0\) and \(x^{2} + y^{2} – 2y – 15 = 0\) is / are

1. 1. (5,8),(-4,3)
2. 2. (8,5)
3. 3. (3,1)
4. 4. (2,1),(4,3)

68. The condition that the lines joining the origin to the points of intersection of the two curves \(x^{2} + y^{2} + gx + c = 0, x^{2} + y^{2} + 2fy – c = 0\) are at right angles, is

1. 1. \(g^{2} – f^{2} = 4c\)
2. 2. \(g^{2} – f^{2} = 2c\)
3. 3. \(f^{2} – 4g^{2} = 8c\)
4. 4. \(g^{2} – 4f^{2} = 8c\)

69. If \(\alpha\) represent the square of the distance between the origin and the point of intersection of the lines \(x^{2} – y^{2} – x + 3y – 2 = 0\) and \(\beta\) represent the product of the perpendicular distances from the origin on the pair of lines, then \(\alpha \beta =\)

1. 1. \(\frac{5}{4}\)
2. 2. 1
3. 3. \(\frac{5}{2}\)
4. 4. 4

70. If the parametric equations of the circle passing through the points (3,4),(3,2) and (1,4) is \(x = a + r \cos \theta , y = b + r \sin \theta\) , then \(b^{2} r^{2} =\)

1. 1. 27
2. 2. 18
3. 3. 9
4. 4. 54

71. From a point P on the circle \(x^{2} + y^{2} – 4x – 6y + 9 = 0\) , a pair of tangents \(PQ\) and \(PR\) are drawn touching the circle \(x^{2} + y^{2} – 4x – 6y + 12 = 0\) , at \(Q\) and \(R\) . If \(C\) is the centre of the concentric circles, then the area of the \(\Delta CQR\) (in sq. units) is

1. 1. \(\frac{1}{2}\)
2. 2. \(\frac{\sqrt{3}}{2}\)
3. 3. \(\frac{\sqrt{3}}{4}\)
4. 4. \(\frac{3}{4}\)

72. Find the equations of the tangents drawn to the circle \(x^{2} + y^{2} = 50\) at the points where the line \(x + 7 = 0\) meets it

1. 1. \(7x + y + 50 = 0 \& 7x – y + 50 = 0\)
2. 2. \(x + y = 0 \& x – y = 0\)
3. 3. \(x + 7y + 5 = 0 \& y – 7x + 5 = 0\)
4. 4. \(x + 7y + 50 = 0 \& x – 7y + 50 = 0\)

73. If the chord of the contact of tangents from a point on the circle \(x^{2} + y^{2} = r_{1}^{2}\) to the circle \(x^{2} + y^{2} = r_{2}^{2}\) touches the circle \(x^{2} + y^{2} = r_{3}^{2}\) then \(r_{1},r_{2},r_{3}\) are in

1. 1. AP
2. 2. HP
3. 3. GP
4. 4. AGP

74. Find the equation of the circle passing through \((1, – 2)\) and touching the \(x\) -axis at \((3,0)\) .

1. 1. \(x^{2} + y^{2} + 6x – 4y – 9 = 0\)
2. 2. \(x^{2} + y^{2} – 6x – 4y + 9 = 0\)
3. 3. \(x^{2} + y^{2} – 6x – 4y – 9 = 0\)
4. 4. \(x^{2} + y^{2} – 6x + 4y + 9 = 0\)

75. Let \(L_{1}\) be a straight line passing through the origin and \(L_{2}\) be the straight line \(x + y = 1\) . If the intercepts made by the circle \(x^{2} + y^{2} – x + 3y = 0\) on \(L_{1}\) and \(L_{2}\) are equal, then which of the following equations represent \(L_{1}\)

1. 1. \(x + y = 0 \& x + 7y = 0\)
2. 2. \(x – y = 0 \& x + 7y = 0\)
3. 3. \(x – 7y = 0 \& x + y = 0\)
4. 4. \(x – 7y = 0 \& x – y = 0\)

76. The equations of the tangents to the circle \(x^{2} + y^{2} = 4\) drawn from the point \((4,0)\) are

1. 1. \(y = \pm \frac{1}{\sqrt{3}} (x – 4)\)
2. 2. \(y = \pm \frac{2}{\sqrt{3}} (x – 4)\)
3. 3. \(x = \pm \frac{1}{\sqrt{3}} (y – 4)\)
4. 4. \(x = \pm \frac{2}{\sqrt{3}} (y – 4)\)

77. If \(P(- 9, – 1)\) is a point on the circle \(x^{2} + y^{2} + 4x + 8y – 38 = 0,\) of the tangent drawn at the other end of the diameter drawn through \(P\) .

1. 1. \(7x – 3y = 60\)
2. 2. \(7x – 3y = 56\)
3. 3. \(7x + 3y = 56\)
4. 4. \(7x + 3y = 60\)

78. Find the equation of a circle whose radius is 5 units and passes through two points on the \(x\) -axis which are at a distance of 4 units from the origin.

1. 1. \(x^{2} + y^{2} – 6x – 25 = 0\)
2. 2. \(x^{2} + y^{2} – 6y – 25 = 0\)
3. 3. \(x^{2} + y^{2} + 6y – 16 = 0\)
4. 4. \(x^{2} + y^{2} + 6x – 16 = 0\)

79. If a foot of the normal from the point \((4,3)\) to a circle is \((2,1)\) and \(2x – y – 2 = 0\) is a diameter of the circle, then the equation of the circle is

1. 1. \(x^{2} + y^{2} + 2x + 1 = 0\)
2. 2. \(x^{2} + y^{2} + 2x – 1 = 0\)
3. 3. \(x^{2} + y^{2} – 2x – 1 = 0\)
4. 4. \(2(x^{2} + y^{2}) – 2x – 1 = 0\)

80. The length of the tangent from any point on the circle \((x – 3)^{2} + (y + 2)^{2} = 5r^{2}\) to the circle \((x – 3)^{2} + (y + 2)^{2} – r^{2}\) is 16 units, then the area between the two circles in sq. units is

1. 1. \(32\pi\)
2. 2. \(4\pi\)
3. 3. \(8\pi\)
4. 4. \(256\pi\)

81. Find the equation of the circle which passes through origin and cuts off the intercepts -2 and 3 over the x and y axes respectively.

1. 1. \(x^{2} + y^{2} – 2x + 8y = 0\)
2. 2. \(2(x^{2} + y^{2}) + 2x – 3y = 0\)
3. 3. \(x^{2} + y^{2} – 2x – 8y = 0\)
4. 4. \(x^{2} + y^{2} + 2x – 3y = 0\)

82. The angle between the pair of tangents drawn from \((1,1)\) to the circle \(x^{2} + y^{2} + 4x + 4y – 1 = 0\)

1. 1. \(\frac{\pi}{2}\)
2. 2. \(\frac{\pi}{4}\)
3. 3. \(\frac{\pi}{3}\)
4. 4. \(\frac{\pi}{6}\)

83. If the circle \(x^{2} + y^{2} – 4x – 8y – 5 = 0\) intersects the line \(3x – 4y – m = 0\) in two distinct points, then the number of integral values of ‘m’ is

1. 1. 52
2. 2. 51
3. 3. 50
4. 4. 49

84. Let \(C\) be the circle with center \((0,0)\) and radius 3 units. The equation of the locus of the mid points of the chords of the circle \(C\) that subtends an angle of \(\frac{2\pi}{3}\) at its center is

1. 1. \(x^{2} + y^{2} = \frac{1}{4}\)
2. 2. \(x^{2} + y^{2} = \frac{27}{4}\)
3. 3. \(x^{2} + y^{2} = \frac{9}{4}\)
4. 4. \(x^{2} + y^{2} = \frac{5}{4}\)

85. The points where the circle \(x^{2} + y^{2} – 3x – 4y + 2 = 0\) cuts the \(x\) -axis are

1. 1. \((1,2)\& (2,0)\)
2. 2. \((2,0)\& (3,0)\)
3. 3. \((0,2)\& (0,1)\)
4. 4. \((1,0)\& (2,0)\)

86. The centre and radius of the circle \(x^{2} + y^{2} + 8x + 10y – 8 = 0\) respectively are and units

1. 1. \((-4, -5),7\)
2. 2. \((4,5),49\)
3. 3. \((-8, -10),8\)
4. 4. \((-4,5),7\)

87. The poles of the tangents to the circle \(x^{2} + y^{2} = 4\) with respect to the circle \((x + 2)^{2} + y^{2} = 8\) , lie on

1. 1. \(y^{2} + 8x = 0\)
2. 2. \(x^{2} + 8y = 0\)
3. 3. \(y^{2} – 8x = 0\)
4. 4. \(x^{2} – 8y = 0\)

88. If the power of the point (1,6) with respect to the circle \(x^{2} + y^{2} + 4x – 6y – a = 0\) is -16 then ‘a’ equals

1. 1. 5
2. 2. 11
3. 3. 21
4. 4. 6

89. The circle \(x^{2} + y^{2} + 4x – 4y + 4 = 0\) touches

1. 1. \(x\) -axis only
2. 2. \(y\) -axis only
3. 3. \(x\) -axis and \(y\) -axis
4. 4. \(x = y\)

90. Given three collinear points \(A(3,1),B(7, – 1)\) and (5,0). The length of a tangent drawn from A to any circle that passes through B and C is units.

1. 1. \(2\sqrt{10}\)
2. 2. \(3\sqrt{10}\)
3. 3. \(\sqrt{10}\)
4. 4. \(\sqrt{20}\)

91. Suppose a circle passes through (2,2) and (9,9) and touches the \(x\) -axis at P. If \(O\) is the origin, then OP is equal to

1. 1. 4
2. 2. 5
3. 3. 6
4. 4. 9

92. The radius of a circle whose centre lies in the fourth quadrant and touches each of the three lines \(x = 0,y = 0\) and \(3x + 4y – 12 = 0\) , is units.

1. 1. 1
2. 2. 2
3. 3. 3
4. 4. 4

93. If the lengths of the tangents drawn from the point (1,2) to the circles \(x^{2} + y^{2} + x + y – 4 = 0\) and \(3x^{2} + 3y^{2} – x – y – \lambda = 0\) are in the ratio 3 : 4. Then \(\lambda =\)

1. 1. \(\frac{23}{4}\)
2. 2. \(\frac{17}{4}\)
3. 3. \(\frac{28}{3}\)
4. 4. \(\frac{19}{4}\)

94. If the length of the tangent drawn from the point from \((-2,3)\) to the circle \(x^{2} + y^{2} + 8x – 6y + k = 0\) is 4 units, then \(k =\)

1. 1. 34
2. 2. 36
3. 3. 38
4. 4. 37

95. If two diameters of a circle o circumference \(10\pi\) lie along the lines \(2x + 3y + 1 = 0\) and \(3x – y – 4 = 0\) , then the equation of circle is

1. 1. \(x^{2} + y^{2} + 2x – 2y – 23 = 0\)
2. 2. \(x^{2} + y^{2} – 2x + 2y – 23 = 0\)
3. 3. \(x^{2} + y^{2} + 2x + 2y – 23 = 0\)
4. 4. \(x^{2} + y^{2} – 2x – 2y – 23 = 0\)

96. The lengths of the tangents from the point (1,2) to the circle \(x^{2} + y^{2} + x + y – 4 = 0\) and \(3x^{2} + 3y^{2} – x – y – k = 0\) are in the ratio 4:3, then the value of \(k\) is

1. 1. \(\frac{9}{4}\)
2. 2. \(\frac{13}{4}\)
3. 3. \(\frac{17}{4}\)
4. 4. \(\frac{21}{4}\)

97. The equation of a circle with center at \((-2,3)\) and circumference of \(4\pi\) units is

1. 1. \(x^{2} + y^{2} + 4x – 6y – 9 = 0\)
2. 2. \(x^{2} + y^{2} + 4x – 6y + 9 = 0\)
3. 3. \(x^{2} + y^{2} + 4x – 6y – 3 = 0\)
4. 4. \(x^{2} + y^{2} – 4x + 6y – 9 = 0\)

98. If a circle of a constant radius 6 passes through origin O and meets the coordinate axes at A and B, then find the locus of the centroid of triangle OAB.

1. 1. \(x^{2} + y^{2} = 4\)
2. 2. \(x^{2} + y^{2} = 36\)
3. 3. \(x^{2} + y^{2} = 16\)
4. 4. \(x^{2} + y^{2} = 6\)

99. If the point \((\lambda ,1 + \lambda)\) lies inside the circle \(x^{2} + y^{2} = 1\) , then

1. 1. \(\lambda > 0\)
2. 2. \(\lambda < 0\)
3. 3. \(-1< \lambda < 0\)
4. 4. \(0< \lambda < 1\)

100. The perpendicular distance from the point (1, 2) to common chord of the circles \(x^{2} + y^{2} – 2x + 4y – 4 = 0\) and \(x^{2} + y^{2} + 4x – 6y – 3 = 0\) is units

1. 1. \(\frac{13}{\sqrt{123}}\)
2. 2. \(\frac{13}{\sqrt{136}}\)
3. 3. \(\frac{13}{\sqrt{63}}\)
4. 4. \(\frac{13}{\sqrt{132}}\)

101. If \((6, – k)\) and \((-3,2)\) are conjugate points with reset to circle \(x^{2} + y^{2} + 6x + 4y + 12 = 0\) then ‘k’ equals

1. 1. \(\frac{-7}{4}\)
2. 2. \(\frac{7}{4}\)
3. 3. \(\frac{4}{7}\)
4. 4. \(\frac{-4}{7}\)

102. If the parametric values of two points A, B on the circle \(x^{2} + y^{2} – 6x + 4y – 12 = 0\) are \(30^{\circ}\) and \(90^{\circ}\) respectively, then the equation of chord AB is

1. 1. \(x + \sqrt{3} y = 0\)
2. 2. \(x – \sqrt{3} y = 0\)
3. 3. \(x + \sqrt{3} y – 3\left(1 + \sqrt{3}\right) = 0\)
4. 4. \(\sqrt{3} x + \sqrt{3} y + 61 = 0\)

103. The length of the chord intercepted by the circle \(x^{2} + y^{2} – 8x – 2y – 8 = 0\) on the line \(x + y + 1 = 0\) is units.

1. 1. 14
2. 2. 7
3. 3. \(2\sqrt{7}\)
4. 4. \(\sqrt{7}\)

104. The length of the chord joining points \((4\cos \theta ,4\sin \theta)\) and \(\left[4\cos (\theta +60^{\circ}),4\sin (\theta +60^{\circ})\right]\) on the circle \(x^{2} + y^{2} = 16\) is

1. 1. 4
2. 2. 8
3. 3. 16
4. 4. 2

105. If the lines \(x + 2y – 5 = 0\) and \(3x – y – 1 = 0\) denote two diameters of a circle of radius 5 units, then the equation of the circle is

1. 1. \(x^{2} + y^{2} – 2x + 4y – 20 = 0\)
2. 2. \(x^{2} + y^{2} – 2x – 4y – 20 = 0\)
3. 3. \(x^{2} + y^{2} + 2x – 4y + 20 = 0\)
4. 4. \(x^{2} + y^{2} + 2x + 4y + 20 = 0\)

106. The equation of polar of (1, 1) with respect to the circle \(x^{2} + y^{2} + 4x + 6y – 3 = 0\) is

1. 1. \(2x + 3y – 1 = 0\)
2. 2. \(3x + 4y + 2 = 0\)
3. 3. \(4x + 3y + 2 = 0\)
4. 4. \(3x + 4y + 4 = 0\)

107. The equation of the circle which touches the \(x\) -axis and \(y\) -axis at the point (1,0) and (0,1) respectively is

1. 1. \(x^{2} + y^{2} – 4y + 3 = 0\)
2. 2. \(x^{2} + y^{2} – 2y + 2 = 0\)
3. 3. \(x^{2} + y^{2} – 2x – 2y + 2 = 0\)
4. 4. \(x^{2} + y^{2} – 2x – 2y + 1 = 0\)

108. The equations of the tangent to the circle \(5x^{2} + 5y^{2} = 1\) parallel to the line \(3x + 4y = 1\) are

1. 1. \(3x + 4y = \pm 2\sqrt{5}\)
2. 2. \(3x + 4y = \pm \sqrt{5}\)
3. 3. \(6x + 8y = \pm \sqrt{5}\)
4. 4. \(3x + 4y = \pm 3\sqrt{5}\)

109. The area of a circle having the lines \(3x – 4y + 4 = 0\) and \(6x – 8y – 7 = 0\) as two of its tangents, is

1. 1. \(\frac{9\pi}{4}\)
2. 2. \(\frac{9\pi}{16}\)
3. 3. \(\frac{3\pi}{4}\)
4. 4. \(\frac{3\pi}{16}\)

110. The polars of (-1,2) with respect to the circles \(S_{1} = x^{2} + y^{2} + 6y + 7 = 0\) and \(S_{2} = x^{2} + y^{2} + 6x + 1 = 0\) are

1. 1. Parallel
2. 2. Coincident
3. 3. perpendicular
4. 4. intersecting at a non zero point

111. The internal centre of similitude of the two circles \(x^{2} + y^{2} – 4x – 6y + 12 = 0\) \(x^{2} + y^{2} + 4x – 2y – 4 = 0\)

112. If a circle has its centre on the line \(x – y – 1 = 0\) and passes through the points of intersection of the two circles \(x^{2} + y^{2} + 2x – 2y – 2 = 0\) and \(x^{2} + y^{2} – 2x + 2y – 7 = 0\) , then the centre of that circle is

1. 1. \(\left(\frac{-1}{2},\frac{-3}{2}\right)\)
2. 2. \(\left(\frac{1}{2},\frac{-1}{2}\right)\)
3. 3. \(\left(\frac{1}{3},\frac{-2}{3}\right)\)
4. 4. \(\left(-2, -3\right)\)

113. If \((a,b)\) is the centre of the circle passing through the vertices of the triangle formed by \(x + y = 6,2x + y = 4\) and \(x + 2y = 5\) , then \((a,b)\) is

1. 1. \((-17, -16)\)
2. 2. \(\left(\frac{17}{2},\frac{19}{2}\right)\)
3. 3. \((17,18)\)
4. 4. \(\left(\frac{-17}{2},\frac{-19}{2}\right)\)

114. The locus of the mid points of the chords of the circle \(x^{2} – 2x + y^{2} = 0\) drawn from a point (0,0) on it is

1. 1. \(x^{2} + y^{2} – x = 0\)
2. 2. \(2x^{2} + y – 2 = 0\)
3. 3. \(y^{2} + x – 1 = 0\)
4. 4. \(y + x^{2} + 2x – 3 = 0\)

115. The number of possible common tangents that can be drawn to the circles \(x^{2} + y^{2} + 4x – 6y – 3 = 0\) \(x^{2} + y^{2} + 4x – 2y + 1 = 0\)

1. 1. 4
2. 2. 3
3. 3. 1
4. 4. 0

116. If \((2,\alpha)\) does not lie outside the circles \(x^{2} + y^{2} = 13\) and \(x^{2} + y^{2} + x – 2y = 14,\) then \(\alpha\) lies in

1. 1. \((-\infty , -3)\cup (4,\infty)\)
2. 2. \([-3,4]\)
3. 3. \((-\infty , -1)\cup (3,\infty)\)
4. 4. \([-2,3]\)

117. The locus of the centre of the circles passing through the origin and cutting off a chord of length 2 units on the line \(x = 1\) is

1. 1. a straight line
2. 2. a circle
3. 3. a parabola
4. 4. an ellipse

118. The number of common tangents that can be drawn to the circles \(x^{2} + y^{2} = 1\) and \(x^{2} + y^{2} – 2x – 6y + 6 = 0\)

1. 1. 4
2. 2. 0
3. 3. 2
4. 4. 1

119. If the line \(y = mx + C\) is a tangent to the circle \(x^{2} + y^{2} = 16\) then \(m =\)

1. 1. \(\pm \frac{1}{4}\sqrt{C – 16}\)
2. 2. \(\pm \frac{1}{4}\sqrt{C^{2} – 16}\)
3. 3. \(\pm \frac{1}{C}\sqrt{C^{2} + 16}\)
4. 4. \(\pm \frac{1}{16} (C^{2} – 16)\)

120. If the points \((2,3)\) and \((K, – 2)\) are conjugate with respect to the circle \(x^{2} + y^{2} – 2x + 4y – 2 = 0\) then \(K =\)

1. 1. 8
2. 2. 6
3. 3. 4
4. 4. 3

121. The number of common tangents that can be drawn to the circles \(x^{2} + y^{2} – 2x – 2y – 23 = 0\) and \(x^{2} + y^{2} – 4x – 4y – 1 = 0\) is

1. 1. 0
2. 2. 1
3. 3. 1
4. 4. 3

122. A point that lies on the common tangent to circles \(x^{2} + y^{2} – 2x + 18y + 78 = 0\) and \(x^{2} + y^{2} + 8x – 6y – 200 = 0\) among the following options is

1. 1. \(\left(0,\frac{139}{12}\right)\)
2. 2. \(\left(\frac{-137}{5},\frac{-1}{6}\right)\)
3. 3. \(\left(31,\frac{-4}{3}\right)\)
4. 4. \(\left(\frac{-2}{5},\frac{-47}{4}\right)\)

123. The length of the common chord of the circles \(x^{2} + y^{2} + 2x + 3y + 1 = 0\) \(x^{2} + y^{2} + 4x + 3y + 2 = 0\)

1. 1. \(\sqrt{2}\)
2. 2. \(\sqrt{2}\)
3. 3. 2
4. 4. 4

124. If \(\alpha ,\beta\) are the roots of \(x^{2} + 2x – 3 = 0\) and \(\gamma ,\delta\) are the roots of \(y^{2} – y + 4 = 0\) , then the equation of the circle having \((\alpha ,\gamma)\) and \((\beta ,\delta)\) as ends of a diameter is

1. 1. \(x^{2} + y^{2} + 4x – 3y + 2 = 0\)
2. 2. \(x^{2} + y^{2} + 2x – y + 1 = 0\)
3. 3. \(x^{2} + y^{2} – 3x + 4y + 1 = 0\)
4. 4. \(x^{2} + y^{2} – 2x + y – 1 = 0\)

125. If the inverse point of (1,1) with respect to the circle \(x^{2} + y^{2} – 4x – 6y + 12 = 0\) is \((h,k)\) then \(h + k =\)

1. 1. \(\frac{22}{5}\)
2. 2. \(\frac{8}{5}\)
3. 3. 2
4. 4. 5

126. If \(h k p q\neq 0\) and the circles \(x^{2} + y^{2} + 2h x + 2k y = 0\) and \(x^{2} + y^{2} + 2p x + 2q y = 0\) touch each other at the origin, then \(h q – p k – \frac{h q}{p k} =\)

1. 1. -1
2. 2. 0
3. 3. 1
4. 4. 2

127. The circle \(x = 5\cos \theta ,y = 5\sin \theta\) is bounded by the rectangle formed by the lines \(x\pm 6 = 0\& y\pm 6 = 0\) . The area of the triangle that lies inside the rectangle which is formed by the tangent at \(P\left(\frac{2\pi}{3}\right)\) to the circle with two of the above given lines is

1. 1. \(x^{2} + y^{2} + 4x – 6y – 9 = 0\)
2. 2. \(x^{2} + y^{2} – 4x + 6y – 4 = 0\)
3. 3. \(48 + \sqrt{3}\)
4. 4. \(\frac{1}{2}\left(\frac{6\sqrt{3} – 4}{\sqrt{3}}\right)^{2}\)

128. If the two circles \(x^{2} + y^{2} – 2x – 6y + 10 – r^{2} = 0\&\) \(x^{2} + y^{2} – 8x + 2y + 8 = 0\) Have a common chord of non- zero length, then

1. 1. \(2< r< 8\)
2. 2. \(0< r< 2\)
3. 3. \(r = 2,8\)
4. 4. \(8< r< 13\)

129. The Locus of centers of the circles, possessing the same area and having \(3x – 4y + 4 = 0\) and \(6x – 8y + 7 = 0\) as their common tangent, is

1. 1. \(12x – 16y – 15 = 0\)
2. 2. \(12x – 16y + 15 = 0\)
3. 3. \(12x – 16y + 15 = 0\)
4. 4. \(3x – 4y – \frac{11}{2} = 0\)

130. For any two nonzero real numbers a and b if this line \(\frac{x}{a} +\frac{y}{b} = 1\) is a tangent to the circle \(x^{2} + y^{2} = 1\) , then which of the following is true?

1. 1. \(\left(\frac{1}{a},\frac{1}{b}\right)\) lies inside the circle
2. 2. \((a,b)\) lies inside the circle
3. 3. \(\left(\frac{1}{a},\frac{1}{b}\right)\) lies on the circle
4. 4. \((a,b)\) lies on the circle

131. The length of the intercept on the line \(4x – 3y – 10 = 0\) \(x^{2} + y^{2} – 2x + 4y – 20 = 0\)

1. 1. 5
2. 2. 2
3. 3. 10
4. 4. 6

132. The pole of the line \(\frac{x}{a} +\frac{y}{b} = 1\) with respect of the circle \(x^{2} + y^{2} = c^{2}\) is

133. If the tangent at the point P on the circle \(x^{2} + y^{2} + 6x + 6y = 2\) meets the straight line \(5x – 2y + 6 = 0\) at a point Q on the y-axis, then the length of PQ is

1. 1. 5
2. 2. 6
3. 3. 4
4. 4. 3

134. For any real number the point \(\left(\frac{8t}{1 + t^2},\frac{4(1 – t^2)}{1 + t^2}\right)\) lies on a / an

1. 1. Circle of radius 2
2. 2. Circle of radius 4
3. 3. Ellipse with 4 as its major axis length
4. 4. Ellipse with 4 as its minor axis length

135. The area of the circle passing through the points \((5,\pm 2)\) ,(1,2) is

1. 1. \(8\pi\)
2. 2. \(4\pi\)
3. 3. \(2\pi\)
4. 4. \(16\pi\)

136. The ratio of the largest and shortest distances from the point \((2, – 7)\) to the circle \(x^{2} + y^{2} – 14x – 10y – 151 = 0\)

1. 1. \(15:13\)
2. 2. \(7:1\)
3. 3. \(3:2\)
4. 4. \(14:1\)

137. A circle has its centre in the first quadrant and passes through (2,3). If this circle makes intercepts of length 3 and 4 respectively on \(x = 2\) and \(y = 3\) , its equation is

1. 1. \(x^{2} + y^{2} + 3x – 5y + 8 = 0\)
2. 2. \(x^{2} + y^{2} – 4x – 6y + 13 = 0\)
3. 3. \(x^{2} + y^{2} – 6x – 8y + 23 = 0\)
4. 4. \(x^{2} + y^{2} – 8x – 9y + 30 = 0\)

138. The radius of the circle having \(3x – 4y + 4 = 0\) and \(6x – 8y – 7 = 0\) as its tangents is

1. 1. \(\frac{3}{2}\)
2. 2. 3
3. 3. 6
4. 4. \(\frac{3}{4}\)

139. A circle is such that \((x – 2)\) \(\cos \theta +(y – 2)\sin \theta = 1\) touches it for all values of \(\theta\) . Then the circle is

1. 1. \(x^{2} + y^{2} – 4x – 4y + 7 = 0\)
2. 2. \(x^{2} + y^{2} + 4x + 4y + 7 = 0\)
3. 3. \(x^{2} + y^{2} – 4x – 4y – 7 = 0\)
4. 4. \(x^{2} + y^{2} + 4x + 4y – 7 = 0\)

140. The least distance of the point \((10,7)\) from the circle \(x^{2} + y^{2} – 4x – 2y – 20 = 0\) is

1. 1. 6
2. 2. 7
3. 3. 4
4. 4. 5

141. Suppose that the \(x\) – coordinates of the points A and B satisfy \(x^{2} + 2x – a^{2} = 0\) and their \(y\) – coordinates satisfy \(y^{2} + 4y – b^{2} = 0\) . Then the equation of the circle with AB as its diameter is

1. 1. \(x^{2} + y^{2} + 2x + 4y – a^{2} – b^{2} = 0\)
2. 2. \(x^{2} + y^{2} + 2x + 4y + a^{2} + b^{2} = 0\)
3. 3. \(x^{2} + y^{2} – 2x – 4y – a^{2} – b^{2} = 0\)
4. 4. \(x^{2} + y^{2} – 2x – 4y + a^{2} + b^{2} = 0\)

142. The circle touching the \(y\) – axis at a distance 4 units from the origin and cutting off an intercept 6 from \(x\) – axis is

1. 1. \(x^{2} + y^{2}\pm 10x – 8y + 16 = 0\)
2. 2. \(x^{2} + y^{2}\pm 5x – 8y + 16 = 0\)
3. 3. \(x^{2} + y^{2}\pm 5x – 2y – 8 = 0\)
4. 4. \(x^{2} + y^{2}\pm 2x – y – 12 = 0\)

143. The set of all points that are at a distance of at least 2 units from \((- 3,0)\) is

1. 1. \(\left\{(x,y)\mid x^{2} + y^{2} + 6x – 7 > 0\right\}\)
2. 2. \(\left\{(x,y)\mid x^{2} + y^{2} + 6x + 5\geq 0\right\}\)
3. 3. \(\left\{(x,y)\mid x^{2} + y^{2} + 6x + 5< 0\right\}\)
4. 4. \(\left\{(x,y)\mid x^{2} + y^{2} + 6x + 7\leq 0\right\}\)

144. The circle touching the coordinate axes with its centre lying on \(x – 2y – 3 = 0\) is

1. 1. \(x^{2} + y^{2} – 2x + 2y + 1 = 0\)
2. 2. \(x^{2} + y^{2} + 2x – 2y + 1 = 0\)
3. 3. \(x^{2} + y^{2} + 6x + 6y – 9 = 0\)
4. 4. \(x^{2} + y^{2} – 6x – 6y + 9 = 0\)

145. Suppose a circle passes through (0, a) and (b, h) having its centre at (c, 0). Then the value of c is

1. 1. \(\frac{b^{2} – a^{2} + h^{2}}{2b}\)
2. 2. \(\frac{b^{2} + a^{2} – h^{2}}{2b}\)
3. 3. \(\frac{b^{2} – a^{2} + h^{2}}{2a}\)
4. 4. \(\frac{b^{2} + a^{2} – h^{2}}{2a}\)

146. For a circle of diameter R, touching \(x^{2} + y^{2} – 4y = 0\) and passing through (4, 5), Which of the following is correct?

1. 1. \(3\leq R\leq 7\)
2. 2. \(R > 7\)

147. The centre of the circle that passes through the point (0, 1) and touches the curve \(y = x^{2}\) at (2, 4) is

148. The slope of the normal to the circle \(x^{2} + y^{2} + 2gx + 2f y + c = 0\) at \((x_{1}, y_{1})\) is

149. The circle possessing \(y\) -axis as its tangent at (0, 2) and passing through \((-1, 0)\) , also passes through

150. Suppose the tangents drawn to the circle \(x^{2} + y^{2} – 6x – 4y – 11 = 0\) from P(1, 8) touch the circle at A and B. Then the centre of the circle passing through P, A and B is

1. 1. \((2, 5)\)
2. 2. \((-2, -5)\)
3. 3. \((-2, 5)\)
4. 4. \((2, -5)\)

151. If the circle \(x^{2} + y^{2} + 2\alpha x + c = 0\) lies completely inside the circle \(x^{2} + y^{2} + 2\beta x + c = 0\) , then which of the following holds?

1. 1. \(\alpha \beta < 0\)
2. 2. \(C < 0\)
3. 3. \(C = 0\)
4. 4. \(\alpha \beta > 0\)

152. If the line \(3x – 4y = 1\) touches the circle \((x – 1)^{2} + (y + 2)^{2} = 4\) at \((\alpha , \beta)\) , the values of \(\alpha\) and \(\beta\) are

1. 1. \(\alpha = \frac{1}{5}, \beta = \frac{-1}{10}\)
2. 2. \(\alpha = \frac{-1}{5}, \beta = \frac{-2}{5}\)
3. 3. \(\alpha = \frac{-2}{5}, \beta = \frac{-11}{20}\)
4. 4. \(\alpha = \frac{2}{5}, \beta = \frac{1}{20}\)

153. The equation of a tangent to the circle \(x^{2} + y^{2} = 1\) , which is perpendicular to the line \(y = mx + 1\) , is

1. 1. \(x + my – \sqrt{1 + m^{2}} = 0\)
2. 2. \(mx + y – \sqrt{1 + m^{2}} = 0\)
3. 3. \(x + my + \sqrt{1 + m^{2}} = 0\)
4. 4. \(mx + y + \sqrt{1 + m^{2}} = 0\)

154. If the equation \(\mathbf{a}\mathbf{x}^{2} + \mathbf{b}\mathbf{y}^{2} + 2\mathbf{h}\mathbf{x}\mathbf{y} + 2\mathbf{g}\mathbf{x} + 2\mathbf{f}\mathbf{y} + \mathbf{c} = 0\) represents a circle passing through the origin, then

1. 1. \(\mathbf{a} = \mathbf{b},\mathbf{c} = 0\)
2. 2. \(\left|a\right| = \left|b\right|,h = 0 = c\)
3. 3. \(\mathbf{a} = \mathbf{b},\mathbf{h} = \mathbf{c} = 0\)
4. 4. \(\mathbf{a} = \mathbf{b},\mathbf{h} = 0\)

155. Suppose \(d_{1}\) and \(d_{2}\) are respectively the lengths of intercepts of the circle \(x^{2} + y^{2} = 4\) and \(x^{2} + y^{2} – 10x – 14y + 65 = 0\) on the line \(2x – 2y – 3 = 0\) Then which of the following is true?

1. 1. \(d_{1} = 2d_{2}\)
2. 2. \(d_{2} = 2d_{1}\)
3. 3. \(d_{1} = 3d_{2}\)
4. 4. \(d_{1} = d_{2}\)

156. If the point \((2,\lambda)\) lies inside the circles \(x^{2} + y^{2} = 13\) and \(x^{2} + y^{2} + x – 2y = 14,\) then \(\lambda\) lies in the set

1. 1. \((-\infty , -3)\cup (4,\infty)\)
2. 2. \((-\infty , -1)\cup (3,\infty)\)
3. 3. \([-3,4]\)
4. 4. \((-2,3)\)

157. For different real non zero numbers \(x_{1},x_{2},x_{3}\) and \(x_{4}\) suppose the points \(\left(x_{1},\frac{1}{x_{1}}\right),\left(x_{2},\frac{1}{x_{2}}\right),\left(x_{3},\frac{1}{x_{3}}\right)\) and \(\left(x_{4},\frac{1}{x_{4}}\right)\) lie on the boundary of a circle of radius 4. Then the value of \(x_{1}x_{2}x_{3}x_{4}\)

1. 1. 1
2. 2. 2
3. 3. 4
4. 4. \(\frac{1}{4}\)

158. The equation of the tangent to the circle \(x^{2} + y^{2} – 9 = 0\) , making an angle \(\theta^{0}\) with the x axis is

1. 1. \(\frac{1}{\sqrt{3}} x – y\pm 6 = 0\)
2. 2. \(\sqrt{3} x – y\pm 6 = 0\)
3. 3. \(\sqrt{3} x + y\pm 6 = 0\)
4. 4. \(\frac{1}{\sqrt{3}} x + y\pm 6 = 0\)

159. The straight line touching the circle \(x^{2} + y^{2} – 2x – 3 = 0\) and remaining normal to the circle \(x^{2} + y^{2} – 4y – 6 = 0\) is

1. 1. \(4x – 3y + 6 = 0\)
2. 2. \(y + 2 = 0\)
3. 3. \(4x + 3y – 6 = 0\)
4. 4. \(2x + 3 = 0\)

160. The ratio of the areas of the greatest and the smallest circles touching \(\left(x\pm 1\right)^{2} + \left(y\pm 1\right)^{2} = 1\) is

1. 1. \(\frac{\sqrt{3} + 1}{\sqrt{3} – 1}\)
2. 2. \(\frac{3 + \sqrt{2}}{3 – \sqrt{2}}\)
3. 3. \(\frac{3 + 2\sqrt{2}}{3 – 2\sqrt{2}}\)
4. 4. 4

161. The equation of circle with centre \((2, – 3)\) and touching x- axis is

1. 1. \(x^{2} + y^{2} – 4x – 6y + 4 = 0\)
2. 2. \(x^{2} + y^{2} – 4x – 6y – 8 = 0\)
3. 3. \(x^{2} + y^{2} – 4x + 6y + 4 = 0\)
4. 4. \(x^{2} + y^{2} + 4x – 6y + 8 = 0\)

162. In a square ABCD of side length a, suppose AB and AD are along the coordinate axes. Then the circle that circumscribes the square is

1. 1. \(x^{2} + y^{2} + a\left(x + y\right) = 0\)
2. 2. \(x^{2} + y^{2} – a\left(x + y\right) = 0\)
3. 3. \(x^{2} + y^{2} + 2a\left(x + y\right) = 0\)
4. 4. \(x^{2} + y^{2} – 2a\left(x + y\right) = 0\)

163. If a point \(P(\alpha ,\beta)\) on the line \(y = 1\) is such that the two distinct chords drawn on \(x^{2} + y^{2} – \alpha x – y = 0\) from P are bisected by the x-axis, then

1. 1. \(\alpha^{2}< 8\)
2. 2. \(\alpha = 2\sqrt{2}\)
3. 3. \(\alpha^{2} > 8\)
4. 4. \(\alpha = -2\sqrt{2}\)

164. The ratio of the areas of the concentric circles \(x^{2} + y^{2} – 6x + 12y + 15 = 0\) and \(x^{2} + y^{2} – 6x + 12y – 15 = 0\) is

1. 1. \(1:\sqrt{2}\)
2. 2. \(1:\sqrt{3}\)
3. 3. \(1:2\)
4. 4. \(1:4\)

165. For any real number \(\lambda \neq 1\) , the centre of the circle that passes through \(A(1,\lambda),B(\lambda ,1)\) and \(C(\lambda ,\lambda)\) is

1. 1. \(\left(\frac{1 + \lambda}{2},\frac{1 + \lambda}{2}\right)\)
2. 2. \(\left(\frac{1 + 2\lambda}{3},\frac{1 + 2\lambda}{3}\right)\)
3. 3. \((1 + 2\lambda ,1 + 2\lambda)\)
4. 4. \(\left(\frac{\lambda}{2},\frac{\lambda}{2}\right)\)

166. The shortest distance from the line \(3x + 4y = 25\) to the circle \(x^{2} + y^{2} – 6x + 8y = 0\) is

1. 1. \(\frac{9}{5}\)
2. 2. \(\frac{7}{5}\)
3. 3. \(\frac{8}{5}\)
4. 4. \(\frac{13}{5}\)

167. If a circle of radius 3 passes through the point (7,3) and has its centre on the line x-y-1=0, then its equation among the following is

1. 1. \(x^{2} + y^{2} + 14x – 12y + 76 = 0\)
2. 2. \(x^{2} + y^{2} + 14x – 12y – 76 = 0\)
3. 3. \(x^{2} + y^{2} + 8x – 6y + 16 = 0\)
4. 4. \(x^{2} + y^{2} – 14x – 12y + 76 = 0\)

168. If the segments of the straight lines \(x + y = 6\) and \(x + 2y = 4\) are two diameters of a circle passing through (6,2), then the equation of that circle is

1. 1. \(x^{2} + y^{2} – 2x – 4y – 20 = 0\)
2. 2. \(x^{2} + y^{2} + 6x – 4y – 68 = 0\)
3. 3. \(x^{2} + y^{2} – 16x + 4y + 48 = 0\)
4. 4. \(x^{2} + y^{2} + 2x – 10y – 32 = 0\)

169. The circle \(x^{2} + y^{2} – 4x – 8y + 16 = 0\) rolls up along the tangent drawn to it at \((2 + \sqrt{3},3)\) by 2 units. The equation of the circle in the new position is

1. 1. \(x^{2} + y^{2} – 6x – 2(4 + \sqrt{3})y + (24 + 8\sqrt{3}) = 0\)
2. 2. \(x^{2} + y^{2} – 6x + 2(4 + \sqrt{3})y + (24 + 8\sqrt{3}) = 0\)
3. 3. \(x^{2} + y^{2} + 6x – 2(4 + \sqrt{3})y + (24 + 8\sqrt{3}) = 0\)
4. 4. \(x^{2} + y^{2} + 6x + 2(4 + \sqrt{3})y + (24 + 8\sqrt{3}) = 0\)

170. Suppose the angle between the tangents drawn from (0,0) to the circle \((x + \lambda)^{2} + (y + 1)^{2} = \lambda^{2}\) is \(\frac{\pi}{2}\) . Then \(\lambda\) satisfies

1. 1. \(\lambda^{2} = 1\)
2. 2. \(\lambda = 0\)
3. 3. \(\lambda^{2} = 4\)
4. 4. \(\lambda^{2} = 9\)

171. A circle passes through the points (1, 2), (3, 4). If its centre lies on the line \(x – y + 3 = 0\) , then its radius is equal to

1. 1. 4
2. 2. 3
3. 3. 1
4. 4. 2

172. A line drawn through the point A(5, 7) cuts the circle \(x^{2} + y^{2} – 36 = 0\) at the points P and Q. Then, AP.AQ =

1. 1. 110
2. 2. 60
3. 3. 38
4. 4. 12

173. Let P be any point on the circle \(x^{2} + y^{2} – 2x – 1 = 0\) and C be its centre. Let AB be the chord of contact of P with respect to the circle \(x^{2} + y^{2} – 2x = 0\) . Then the locus of the circumcentre of the triangle CAB is

1. 1. \(2x^{2} + 2y^{2} – 4x + 1 = 0\)
2. 2. \(x^{2} + y^{2} – 4x + 2 = 0\)
3. 3. \(x^{2} + y^{2} – 4x + 1 = 0\)
4. 4. \(2x^{2} + 2y^{2} – 4x + 3 = 0\)

174. If a circle C passing through (4,0) touches the circle \(x^{2} + y^{2} + 4x – 6y – 12 = 0\) externally at the point (1, -1), then the radius of C is

1. 1. \(\sqrt{12}\)
2. 2. 4
3. 3. \(\sqrt{3}\)
4. 4. 5

175. If the circles \(C_{1}:x^{2} + y^{2} + 2x + 4y – 20 = 0,\) \(C_{2}:x^{2} + y^{2} + 6x – 8y + 9 = 0\) have \(n\) common tangents and the length of the tangent drawn from the centre of similitude to the circle \(C_{2}\) is \(l\) then \(\frac{l}{n^{2}} =\)

1. 1. \(4\sqrt{39}\)
2. 2. \(\sqrt{39}\)
3. 3. \(\frac{\sqrt{39}}{4}\)
4. 4. \(2\sqrt{39}\)

176. From a point \(A(0,3)\) on the circle \((x + 2)^{2} + (y – 3)^{2} = 4\) a chord AB is drawn and it is extended to a point Q such that \(AQ = 2AB\) . Then the locus of Q is

1. 1. \((x + 4)^{2} + (y – 3)^{2} = 16\)
2. 2. \((x + 1)^{2} + (y – 3)^{2} = 32\)
3. 3. \((x + 1)^{2} + (y – 3)^{2} = 4\)
4. 4. \((x + 1)^{2} + (y – 3)^{2} = 1\)

177. If \(m_{1},m_{2}\) are the slopes of the tangents drawn from a point (1, -3) to the circle \(x^{2} + y^{2} – 6x + 4y + 12 = 0\)

1. 1. 16
2. 2. 25
3. 3. 4
4. 4. 1

178. If A, B are the points of contact of the tangents drawn from the point P (-2, -3) to the circle \(x^{2} + y^{2} – 8x – 10y + 5 = 0\) and the chord AB subtends an angle \(\theta\) at P then \(\tan \theta =\)

1. 1. \(\frac{3}{4}\)
2. 2. \(\frac{24}{7}\)
3. 3. \(\frac{7}{24}\)
4. 4. \(\frac{4}{3}\)

179. The equation of the transverse common tangent of the circles \(x^{2} + y^{2} – 6x – 8y + 9 = 0\) and \(x^{2} + y^{2} + 2x – 2y + 1 = 0\)

1. 1. \(4x + 3y – 4 = 0\)
2. 2. \(3x + y – 1 = 0\)
3. 3. \(2x – y + 2 = 0\)
4. 4. \(x + 2y – 3 = 0\)

180. If \(\theta\) is the angle between the circles \(x^{2} + y^{2} – 2x – 4y – 4 = 0\) \(x^{2} + y^{2} – 8x – 12y + 43 = 0\) \(7\sec \theta -18\cos \theta =\)

1. 1. 11
2. 2. 9
3. 3. 0
4. 4. 1

181. The equation of the incircle of the triangle formed by the lines \(x = 0,y = 0\) and \(3x + 4y – 24 = 0\)

1. 1. \(x^{2} + y^{2} – 24x – 24y + 144 = 0\)
2. 2. \(x^{2} + y^{2} – 6x – 6y + 9 = 0\)
3. 3. \(x^{2} + y^{2} – 4x – 4y + 4 = 0\)
4. 4. \(x^{2} + y^{2} – 8x – 8y + 16 = 0\)

182. If two tangents are drawn from the point \(P\left(\frac{\pi}{4}\right)\) on the circle \(x^{2} + y^{2} = 4\) to the circle \(x^{2} + y^{2} = 1\) then the slopes of the tangents are

1. 1. \(2\pm \sqrt{2}\)
2. 2. \(1\pm \sqrt{2}\)
3. 3. \(2\pm \sqrt{3}\)
4. 4. \(1\pm \sqrt{3}\)

183. If \(5x + 6y – 34 = 0\) and \(2x + y + c = 0\) are conjugate lines with respect to the circle \(x^{2} + y^{2} – 8x – 10y + 25 = 0\) then the point on the line \(2x + y + c = 0\) is

1. 1. (3,3)
2. 2. (2,4)
3. 3. (1,-5)
4. 4. (-2,-2)

184. If \(C_{1}\) and \(C_{2}\) are the centres of similitude with respect to the circles \(x^{2} + y^{2} + 6x + 8y + 24 = 0\) and \(x^{2} + y^{2} – 6x – 8y + 9 = 0\) then \(C_{1}C_{2} =\)

1. 1. 10
2. 2. 5
3. 3. \(\frac{16}{3}\)
4. 4. \(\frac{19}{3}\)

185. The radius of the circle passing through the points \((-1,1)\) , \((2, – 1)\) and \((1,0)\) is

1. 1. 5
2. 2. \(\frac{\sqrt{130}}{2}\)
3. 3. 6
4. 4. \(\frac{\sqrt{145}}{2}\)

186. If \(A = (0, – 2)\) and \(B\) is any point on the circle \(x^{2} + y^{2} – 2x – 2y + 1 = 0\) , then the maximum value of \(\left(\overline{AB}\right)^{2}\) is

1. 1. 51
2. 2. \(11 + 2\sqrt{10}\)
3. 3. \(9 + 3\sqrt{5}\)
4. 4. \(\frac{5 + 2\sqrt{3}}{2}\)

187. If \((\alpha ,\beta)\) is the pole of the line \(3x – 5y + 6 = 0\) with respect to the circle \(x^{2} + y^{2} – 10x + 14y + 46 = 0\) then \(\alpha +\beta =\)

1. 1. -1
2. 2. 8
3. 3. 3
4. 4. -4

188. O(0,0) and A(1,0) are centres of two unit circles \(C_{1}\) and \(C_{2}\) respectively. \(C_{3}\) is also a unit circle having its centre above \(X\) -axis and passing through O and A. The equation of the common tangent to \(C_{1}\) and \(C_{3}\) which does not intersect the circle \(C_{2}\) is

1. 1. \(\sqrt{3} x – y + 2 = 0\)
2. 2. \(x + \sqrt{3} y + 2 = 0\)
3. 3. \(\sqrt{3} x – y – 2 = 0\)
4. 4. \(x + \sqrt{3} y – 2 = 0\)

189. If the circles \(x^{2} + y^{2} – 16x – 20y + 164 = r^{2}(r > 0)\) and \(x^{2} + y^{2} – 8x – 14y + 29 = 0\) intersect in two distinct points, then the maximum possible integral value of \(r\) is

1. 1. 1
2. 2. 10
3. 3. -2
4. 4. 2

190. Let the centre of the circle \(S = 0\) lie on the line \(x + y – 5 = 0\) and also lie in the first quadrant. If this circle touches both the lines \(x – 2 = 0\) and \(y – 5 = 0\) , then the area of the circle is

1. 1. \(\pi\) sq. units
2. 2. \(2\pi\) sq.units
3. 3. \(4\pi\) sq.units
4. 4. \(\frac{1}{4}\pi\) sq.units

191. The straight line \(x + 2y = 1\) cuts the \(X\) -axis at A and \(Y\) -axis at B. A circle is drawn through A, B and the origin. The sum of the perpendicular distances from A, B on to the tangent drawn at origin to the circle S is

1. 1. equal to the radius of the circle S
2. 2. equal to the diameter of the circle S
3. 3. equal to twice the diameter of the circle S
4. 4. equal to \(\sqrt{5}\) times the radius of the circle S

192. Let P and Q be two external points of the circle \(S = x^{2} + y^{2} – a^{2} = 0\) . Let the chord of contact of the point P with respect to the circle \(\mathbf{S} = 0\) pass through Q. If \(l_{1}\) and \(l_{2}\) are the lengths of the tangents drawn from P and Q to the circle \(\mathbf{S} = 0\) , then \(\mathrm{PQ} =\)

1. 1. \(\sqrt{l_{1} + l_{2}}\)
2. 2. \(\frac{l_{1} + l_{2}}{2}\)
3. 3. \(\sqrt{l_{1}^{2} + l_{2}^{2}}\)
4. 4. \(\sqrt{l_{1}^{2} – 2l_{1} + l_{2}^{2} – 2l_{2}}\)

193. A(x1,y1) is the internal centre of similitude and \(\mathbf{B}(\mathbf{x}_{2},\mathbf{y}_{2})\) is the external centre of similitude of two circles \(\mathbf{C}_{1}\) and \(\mathbf{C}_{2}\) whose centres are \(P(\alpha ,\beta)\) and \(Q(\gamma ,\delta)\) respectively. If \(\mathrm{PA} = 3\) \(\mathrm{AB} = 5\) \(\mathrm{QB} = 2\) , then ratio of the radii of the two circles is

1. 1. 2:3
2. 2. 3:2
3. 3. 1:1
4. 4. 5:2

194. The equation of the direct common tangent of the circles \(x^{2} + y^{2} – 6x – 4y – 23 = 0\) and \(x^{2} + y^{2} + 2x + 2y + 1 = 0\) is

1. 1. \(6x – 4y + 1 = 0\)
2. 2. \(3x – 4y + 6 = 0\)
3. 3. \(4x + 3y + 12 = 0\)
4. 4. \(2x – 4y + 3 = 0\)

195. The line \(x + 2y – c = 0\) meets the curve \(x^{2} + y^{2} – 3x – 6y + 3 = 0\) at two points P and Q and \(\left|P O Q = \frac{\pi}{2}\right.\) , where O is the origin. Then, \(2c^{2} – 15c =\)

1. 1. 15
2. 2. -15
3. 3. 2
4. 4. -2

196. The line \(4x + 3y – 4 = 0\) divides the circumference of a circle in the ratio 1 : 2. If C(5, 3) is the centre of that circle, then equation of the circle is

1. 1. \((x – 5)^{2} + (y – 3)^{2} = 10^{2}\)
2. 2. \((x – 5)^{2} + (y – 3)^{2} = 12^{2}\)
3. 3. \((x – 5)^{2} + (y – 3)^{2} = 7^{2}\)
4. 4. \((x – 5)^{2} + (y – 3)^{2} = 8^{2}\)

197. Two sides of a square are along the lines \(x = -5\) and \(y = 4\) . The point of intersection of the diagonals is \((3, -4)\) . The point of intersection of the tangents drawn to the circumcircle of the square at the two consecutive vertices lying on \(x = -5\) is

1. 1. \((-4, -4)\)
2. 2. \((-13, -4)\)
3. 3. \((-4, -13)\)
4. 4. \((-4, -10)\)

198. If \(\mathrm{L}_{1}\) \(\mathrm{L}_{2}\) and \(\mathrm{L}_{3}\) are the chords of contact of the three points (2, 0), (1, -2) and (4, 4) respectively with respect to the circle \(x^{2} + y^{2} = 3\) , then \(\mathrm{L}_{1}\) \(\mathrm{L}_{2}\) \(\mathrm{L}_{3}\) are

1. 1. Concurrent lines
2. 2. Sides of a right-angled triangle
3. 3. Sides of an equilateral triangle
4. 4. Parallel lines

199. The combined equation of the direct common tangents of the circles \(x^{2} + y^{2} + 2x = 0\) and \(x^{2} + y^{2} – 2y – 3 = 0\) is

1. 1. \(xy + x + 2y + 2 = 0\)
2. 2. \(x^{2} – xy – 2y^{2} + 3x – 6y = 0\)
3. 3. \(2x^{2} + 5xy + 2y^{2} + 13x + 14y + 20 = 0\)
4. 4. \(2x^{2} – 9xy + 9y^{2} + 3x – 6y + 1 = 0\)

200. Suppose two tangents PA and PB are drawn to the circle centered at C(1, 2) from the point P(16, 7). If the area of the quadrilateral PACB is 75 square units, then the radius of the circle is

1. 1. 5
2. 2. 25
3. 3. 225
4. 4. \(\sqrt{5}\)

201. If the coordinates of point of contact of the circles \(x^{2} + y^{2} – 4x + 8y + 4 = 0\) and \(x^{2} + y^{2} + 2x = 0\) is \(\left(a,b\right)\) then \(a + 2b =\)

1. 1. -1
2. 2. -2
3. 3. 0
4. 4. 1

202. If the chord of contact of the point P(h, k) with respect to the circle \(x^{2} + y^{2} – 4x – 4y + 8 = 0\) meets the circle in two distinct points and it also makes an angle \(45^{\circ}\) with the positive X- axis in the positive direction, then (h, k) cannot be

1. 1. \(\left(\frac{5}{2},\frac{3}{2}\right)\)
2. 2. \(\left(\frac{5}{3},\frac{7}{3}\right)\)
3. 3. (3,1)
4. 4. (2,2)

203. The equation of the pair of tangents drawn from the point (1, 1) to the circle \(x^{2} + y^{2} + 2x + 2y + 1 = 0\) is

1. 1. \(3x^{2} – 8xy + 3y^{2} – 2x – 2y + 6 = 0\)
2. 2. \(11x^{2} – 8xy + 11y^{2} – 4x – 4y – 6 = 0\)
3. 3. \(3x^{2} – 8xy + 3y^{2} + 2x + 2y – 2 = 0\)
4. 4. \(x^{2} – 4xy + y^{2} + x + y = 0\)

204. The distance between the centres of similitude of the circles \(x^{2} + y^{2} + 6x – 8y + 16 = 0\) and \(x^{2} + y^{2} – 2x – 2y + 1 = 0\) is

1. 1. \(\frac{15}{4}\)
2. 2. \(\frac{5}{4}\)
3. 3. \(\frac{5}{2}\)
4. 4. \(\frac{15}{2}\)

205. Let P and Q be the inverse points with respect to the circle \(S\equiv x^{2} + y^{2} – 4x – 6y + k = 0\) and C be the Centre of the circle \(\mathrm{S} = 0\) such that CP.CQ=4. If \(P = (1,2)\) and \(Q = (a,b)\) , then \(2a =\)

1. 1. b
2. 2. -1
3. 3. 3b
4. 4. 0

206. Let A(2,3), B(3,-1) and C(-3,2) be three points. If the centre of the circle passing through A,B and C is (h, k), then \(2k – 4h =\)

1. 1. 0
2. 2. 2
3. 3. -1
4. 4. 1

207. If \(P\left(\frac{\pi}{3}\right)\) and \(Q\left(\frac{2\pi}{3}\right)\) represent two points on the circle \(x^{2} + y^{2} – 4x + 6y – 12 = 0\) in parametric form, then the length of the chord PQ is

1. 1. \(4\sqrt{3}\)
2. 2. 5
3. 3. \(5\sqrt{2}\)
4. 4. 13

208. If the circles \(x^{2} + y^{2} – 2x + 4y + c = 0\) and \(x^{2} + y^{2} + 2x – 4y + c = 0\) Have four common tangents, then

1. 1. \(c< 0\)
2. 2. \(-2< c< 2\)
3. 3. \(0< c< 5\)
4. 4. \(c > 0\)

209. The locus of the poles of the tangents to the circle \(x^{2} + y^{2} – 2x + 2y – 2 = 0\) with respect to the circle \(x^{2} + y^{2} = 4\) , is

1. 1. \(3x^{2} + 2xy + 3y^{2} + 8x – 8y – 16 = 0\)
2. 2. \(x^{2} – 2xy + y^{2} + 4x + 4y + 8 = 0\)
3. 3. \(3x^{2} + 2xy + 3y^{2} + 4x + 4y + 16 = 0\)
4. 4. \(x^{2} + y^{2} – 4x + 4y – 8 = 0\)

210. Let the circle S which is concentric with the circle \(x^{2} + y^{2} – 2x + ky + 4 = 0\) pass through the point (3,- 2). If one of the diameters of S lies along the line \(3x – 2y + 4 = 0\) , then the radius of the circle S is

1. 1. \(\frac{\sqrt{149}}{2}\)
2. 2. \(\sqrt{31}\)
3. 3. \(\sqrt{38}\)
4. 4. \(\frac{1}{2}\sqrt{137}\)

211. If the length of the chord \(2x + 3y + k = 0\) of the circle \(x^{2} + y^{2} – 6x – 8y + 9 = 0\) is \(2\sqrt{3}\) , then one of the values of \(k\) is

1. 1. 31
2. 2. 5
3. 3. -5
4. 4. -13

212. If \(Q\) is the inverse point of the point \(P(2,3)\) with respect to the circle \(x^{2} + y^{2} – 2x – 2y + 1 = 0\) , then the circle with \(PQ\) as diameter is

1. 1. \(3x^{2} + 3y^{2} – 14x – 16y + 37 = 0\)
2. 2. \(x^{2} + y^{2} – 4x – 6y + 13 = 0\)
3. 3. \(5x^{2} + 5y^{2} – 16x – 22y + 33 = 0\)
4. 4. \(2x^{2} + 2y^{2} – 3x – 3y – 11 = 0\)

213. The lengths of the intercepts made by a circle \(2\sqrt{13}\) \(2\sqrt{22}\) S on X and Y- axes are 3 and 3 respectively. If the radius of the circle S is \(\frac{\sqrt{38}}{3}\) and its centre C lies in the second quadrant, then \(C =\)

1. 1. \(\left(\frac{-5}{3}, \frac{4}{3}\right)\)
2. 2. \(\left(\frac{-4}{3}, \frac{5}{3}\right)\)
3. 3. \(\left(\frac{-6}{5}, \frac{7}{5}\right)\)
4. 4. \(\left(\frac{-7}{5}, \frac{6}{5}\right)\)

214. If the mid point of the chord intercepted by the circle \(x^{2} + y^{2} – 8xy + 10y + 5 = 0\) on the line \(2x + y + 2 = 0\) is (h,k) then \(k + 4h =\)

1. 1. 2
2. 2. 0
3. 3. 1
4. 4. -1

215. If a circle S passing through the points A(1,2) and B(2,1) has its centre C located in the third quadrant at a distance of \(\frac{7}{\sqrt{2}}\) units from AB, then the point P(1,- 2)

1. 1. Lies inside the circle S
2. 2. Lies outside the circle S
3. 3. Lies on the circle S
4. 4. Lies on the line AB

216. The equation of a tangent to the circle \(x^{2} + y^{2} + 2x – 12y – 132 = 0\) which is perpendicular to the line \(12x + 5y + k = 0\) is

1. 1. \(5x – 12y + 92 = 0\)
2. 2. \(5x – 12y – 246 = 0\)
3. 3. \(5x – 12y – 169 = 0\)
4. 4. \(5x – 12y + 246 = 0\)

217. A circle S touches Y-axis at (0, 3) and makes an intercept of length 8 units on X-axis. If the centre C of the circle S lies in the second quadrant, then the distance of C from the point (-2, -1) is

1. 1. 13
2. 2. 10
3. 3. 5
4. 4. \(\sqrt{2}\)

218. If the equation of the circle of radius 3 units which touches the circle \(x^{2} + y^{2} + 6x – 8y – 11 = 0\) externally at (3,0) is \(x^{2} + y^{2} + 2gx + 2fy + c = 0\) , then \(3g – 4f + c =\)

1. 1. 0
2. 2. 5
3. 3. 1
4. 4. -1

219. Tangent \(L_{1} \equiv 3x – 4y – 8 = 0\) and the chord \(L_{2} \equiv x + y – 1 = 0\) are at a distance of 2 and \(\sqrt{2}\) units respectively from the centre of a circle S. (h,k) is the centre of S such that \(h^{2} + k^{2} = 13\) . If the midpoint of the chord \(L_{2} = 0\) is \((\alpha , \beta)\) and the radius of the circle is r, then \(\alpha + \beta + r =\)

1. 1. 4
2. 2. -1
3. 3. 7
4. 4. 3

220. The polar of a point with respect to the circle \(x^{2} + y^{2} – 10x + 12y – 3 = 0\) which is not a tangent and not a chord of contact is

1. 1. \(2x + 3y + 8 = 0\)
2. 2. \(3x + 4y + 5 = 0\)
3. 3. \(5x – 12y + 7 = 0\)
4. 4. \(6x – 8y + 15 = 0\)

221. Let the locus of the point of intersection of the perpendicular tangents drawn to the circle \(x^{2} + y^{2} + 6x – 4y – 12 = 0\) be the circle S. Then the equation of the tangent drawn to S which is perpendicular to the line \(6x – 4y + k = 0\) is

1. 1. \(4x + 6y \pm \sqrt{26} = 0\)
2. 2. \(2x + 3y \pm \sqrt{26} = 0\)
3. 3. \(2x + 3y \pm 5\sqrt{26} = 0\)
4. 4. \(4x + 6y \pm 5\sqrt{26} = 0\)

222. The distance of the origin from the external centre of similitude for the circles \(x^{2} + y^{2} – 8x – 10y – 8 = 0\) \(x^{2} + y^{2} + 2x – 2y – 2 = 0\)

1. 1. \(\frac{3\sqrt{26}}{5}\)
2. 2. \(\frac{\sqrt{290}}{9}\)
3. 3. \(\frac{\sqrt{290}}{5}\)
4. 4. \(\frac{\sqrt{26}}{3}\)

223. Let the equation \(\alpha x^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0\) represent a point circle other than the origin. Then which one of the following conditions must hold?

1. 1. b c>0
2. 2. b>0 and c>0
3. 3. b<0 and c>0
4. 4. b≤0 and c<0

224. The point of intersection of the tangents drawn at the points where the line \(2x – y + 3 = 0\) meets the circle \(x^{2} + y^{2} – 4x – 6y + 4 = 0\) is

1. 1. \((-8, \frac{15}{2})\)
2. 2. \((-5, \frac{21}{4})\)
3. 3. \(\left(\frac{5}{2}, -\frac{21}{4}\right)\)
4. 4. \(\left(8, -\frac{15}{2}\right)\)

225. If \(S \equiv 2x^{2} + 2y^{2} – 8x + 8y – 7 = 0\) is the circle passing through the points of intersection of the circles \(x^{2} + y^{2} + kx – ky + 1 = 0\) and \(x^{2} + y^{2} – kx + ky – 2 = 0\) , then the length of the tangent drawn from the point (k, k) to the circles is

1. 1. \(\sqrt{\frac{29}{2}}\)
2. 2. 3
3. 3. \(\sqrt{\frac{23}{2}}\)
4. 4. \(\sqrt{23}\)

226. The line \(3x + y – 5 = 0\) touches a circle S at (1, 2). If (h, k) is the centre of the circle S such that \(\mathrm{h}^{2} + \mathrm{hk} + \mathrm{k}^{2} = 37\) and the radius of the circle S is \(\sqrt{10}\) , then \(\mathrm{k} =\)

1. 1. 4
2. 2. 3
3. 3. 2
4. 4. 1

227. If \(x + y – 1 = 0\) and \(2x – y + 1 = 0\) are conjugate lines with respect to a circle \(x^{2} + y^{2} – 4x + 2fy – 1 = 0\) , then \(\mathrm{f} =\)

1. 1. -1 or 3
2. 2. 1 or 2
3. 3. -2 or 0
4. 4. -1 or 2

228. The product of the slopes of the common tangents drawn to the circles \(x^{2} + y^{2} + 2x – 2y – 2 = 0\) and \(x^{2} + y^{2} – 2x + 2y + 1 = 0\) which passes through the point (3, -3)

1. 1. -1
2. 2. 3
3. 3. -8
4. 4. 1

229. The length of the chord of contact of the point (2, 1) with respect to the circle \(x^{2} + y^{2} + 4x + 2y + 1 = 0\)

1. 1. \(\frac{8}{\sqrt{5}}\)
2. 2. \(\frac{4}{\sqrt{5}}\)
3. 3. \(\frac{4\sqrt{6}}{\sqrt{5}}\)
4. 4. \(\frac{2\sqrt{6}}{\sqrt{5}}\)

230. Let \(\mathrm{S} = 0\) be the circle passing through the points (2,0),(1,-2),(-1,1). Then the point (1,2)

1. 1. Lies inside the circle \(\mathrm{S} = 0\)
2. 2. Lies outside the circle \(\mathrm{S} = 0\)
3. 3. Lies on the circle \(\mathrm{S} = 0\)
4. 4. is the centre of the circle \(\mathrm{S} = 0\)

231. If the acute angle between the pair of tangents drawn from the origin to the circle \(x^{2} + y^{2} – 4x – 8y + 4 = 0\) is \(\alpha\) , then \(\tan \alpha =\)

1. 1. 3/5
2. 2. 3/4
3. 3. 4/3
4. 4. 4/5

232. Let C be the centre and A be one end of a diameter of the circle \(x^{2} + y^{2} – 2x – 4y – 20 = 0\) . If P is point on AC such that A divides CP in the ratio 2:3, then the locus of P is

1. 1. \(x^{2} + y^{2} – 2x – 4y – 205 = 0\)
2. 2. \(2x^{2} + 2y^{2} – 4x – 8y – 405 = 0\)
3. 3. \(x^{2} + y^{2} – 2x – 4y – 450 = 0\)
4. 4. \(4x^{2} + 4y^{2} – 8x – 16y – 605 = 0\)

233. If the chord of contact of the point P(1, 1) with respect to the circle \(\mathrm{S} = x^{2} + y^{2} + 4x + 6y – 3 = 0\) meet the circle \(\mathrm{S} = 0\) at A and B, then the area of \(\Delta \mathrm{PAB}\) is

1. 1. \(\frac{216}{25}\)
2. 2. \(\frac{108}{25}\)
3. 3. \(\frac{27}{25}\)
4. 4. \(\frac{54}{5}\)

234. Let \(M\left(\frac{-7}{2},\frac{-5}{2}\right)\) be the midpoint of the chord AB of the circle \(x^{2} + y^{2} + 10x + 8y – 23 = 0\) . If \(ax + by + 1 = 0\) is the equation of AB, then \(3a + 3b =\)

1. 1. 6
2. 2. 1
3. 3. 36
4. 4. -1

235. If the inverse point of the point (3, 2) with respect to the circle \(x^{2} + y^{2} – 2x + 4y – 4 = 0\) is \((\ell ,m)\) then \(2\ell +19m =\)

1. 1. 3
2. 2. 1
3. 3. 0
4. 4. -1

236. Let S be a circle concentric with the circle \(3x^{2} + 3y^{2} + x + y – 1 = 0\) . If the length of the tangent drawn from a point (2, -2)to the given circle is the radius of the circle S, then the power of the point (2, 1) with respect to circle S is

1. 1. -137
2. 2. 1
3. 3. -29
4. 4. 23

237. If P (2,3) and Q(-1,2) are conjugate with respect to the circle \(x^{2} + y^{2} + 2gx + 3y – 2 = 0\) Then the radius of the circle is

1. 1. \(\frac{19}{6}\)
2. 2. \(\frac{3\sqrt{21}}{\sqrt{2}}\)
3. 3. \(\frac{3\sqrt{3}}{\sqrt{2}}\)
4. 4. \(\frac{35}{2}\)

238. If a diameter of the circle \(x^{2} + y^{2} – 4x + 6y – 12 = 0\) is a chord of a circle S whose centre is at \((- 3,2)\) , then the radius of S is

1. 1. \(5\sqrt{3}\)
2. 2. \(4\sqrt{3}\)
3. 3. \(2\sqrt{3}\)
4. 4. 5

239. If a circle passing through A(1,1) touches the X-axis, then the locus of the other end of the diameter through A is

1. 1. \((x + 1)^{2} = 4y\)
2. 2. \((y – 1)^{2} = 4x\)
3. 3. \((x – 1)^{2} = 4y\)
4. 4. \((y + 1)^{2} = 4x\)

240. If \(C(\alpha ,\beta)\left(\alpha < 0\right)\) is the centre of the circle that touches the Y-axis at (0,3) and makes an intercept of length 2 units on positive X-axis, then \((\alpha ,\beta) =\)

1. 1. \((-3,\sqrt{10})\)
2. 2. \((-3, – \sqrt{10})\)
3. 3. \((- \sqrt{10},3)\)
4. 4. \((- \sqrt{10}, – 3)\)

241. The equations of the tangents to the circle \(x^{2} + y^{2} = 4\) drawn from the point (4,0) are

1. 1. \(\sqrt{3} y = \pm (x – 4)\)
2. 2. \(\sqrt{3} y = \pm 2(x – 4)\)
3. 3. \(\sqrt{3} x = \pm (y – 4)\)
4. 4. \(\sqrt{3} x = \pm 2(y – 4)\)

242. The image of every point lying on the curve \(x^{2} + y^{2} = 1\) in the line \(x + y = 1\) satisfies the equation

1. 1. \(x^{2} + y^{2} + 2x + 2y + 1 = 0\)
2. 2. \(x^{2} + y^{2} – 2x + 2y + 1 = 0\)
3. 3. \(x^{2} + y^{2} + 2x – 2y + 1 = 0\)
4. 4. \(x^{2} + y^{2} – 2x – 2y + 1 = 0\)

243. If the inverse of \(P(-3,5)\) with respect to a circle is (1,3), then polar of P with respect to that circle is

1. 1. \(x + 2y = 7\)
2. 2. \(2x – 2y + 4 = 0\)
3. 3. \(2x – y + 1 = 0\)
4. 4. \(2x + y – 5 = 0\)

244. If the tangent drawn at the point P on the circle \(x^{2} + y^{2} + 6x + 6y = 2\) meets the straight line \(5x – 2y + 6 = 0\) at a point Q on the Y-axis, then the length of PQ is

1. 1. 5
2. 2. 4
3. 3. 2
4. 4. 1

245. If a circle passing trough (1,-2) has \(x – y = 2\) and \(2x + 3y = 14\) as its diameters, then the radius of the circle is

1. 1. 2
2. 2. 3
3. 3. 4
4. 4. 5

246. The number of common tangents to the circles \(x^{2} + y^{2} – 2x – 6y + 9 = 0\) and \(x^{2} + y^{2} + 6x – 2y + 1 = 0\) is

1. 1. 1
2. 2. 2
3. 3. 3
4. 4. 4

247. The pole of the straight line \(9x + y – 28 = 0\) with respect to the circle \(2x^{2} + 2y^{2} – 3x + 5y – 7 = 0\)

1. 1. (3,1)
2. 2. (-3,1)
3. 3. (-2,1)
4. 4. (3,-1)

248. Let a chord AB subtend an angle of \(60^{\circ}\) at the centre C(2,3) of a circle S. If the equation of AB is \(x + y + 1 = 0\) , then the equation of the circle S is

1. 1. \(x^{2} + y^{2} – 4x – 6y + 11 = 0\)
2. 2. \(x^{2} + y^{2} – 4x – 6y + 37 = 0\)
3. 3. \(x^{2} + y^{2} – 4x – 6y – 11 = 0\)
4. 4. \(x^{2} + y^{2} – 4x – 6y – 37 = 0\)

249. Let 6,8 be the X and Y- intercepts made by the circle \(S\equiv x^{2} + y^{2} + 2gx + 2fy + c = 0\) respectively. If gx+fy+1=0 is a line passing through the point(1,-1),then the radius of the circle \(\mathbf{S} = 0\) is

1. 1. \(\sqrt{41}\)
2. 2. 13
3. 3. \(\sqrt{26}\)
4. 4. 5

250. If(3,1) and (-2,4) are points on a circle S whose centre lies on the line \(x – y + 1 = 0\) then the parametric equations of S are

1. 1. \(x = -1 + \sqrt{17}\cos \theta ,y = \sqrt{17}\sin \theta\)
2. 2. \(x = 2 + \sqrt{13}\cos \theta ,y = 1 + \sqrt{13}\sin \theta\)
3. 3. \(x = \sqrt{26}\cos \theta ,y = -1 + \sqrt{26}\sin \theta\)
4. 4. \(x = -1 + \sqrt{19}\cos \theta ,y = 2 + \sqrt{19}\sin \theta\)

251. Let \(S\equiv x^{2} + y^{2} – 8x + 10y + 5 = 0\) be a circle. Let P(1,1) and Q(1,-1) be two points. Then the point of intersection of the polar of P with respect to \(S = 0\) and the chord with Q as mid- point to \(S = 0\) is

1. 1. (2,2)
2. 2. (11,13/2)
3. 3. (-4,-1)
4. 4. (5,7/2)

252. If the parametric equations of the circle passing through the points (3,4),(3,2) and (1,4)is \(x = a + r\cos \theta ,y = b + r\sin \theta\) then \(b^{r}r^{a} =\)

1. 1. 9
2. 2. 18
3. 3. 27
4. 4. 54

253. A tangent PT is drawn to the circle \(x^{2} + y^{2} = 4\) at the point \(P\left(\sqrt{3},1\right)\) . If a straight line L which is perpendicular to PT is a tangent to the circle \((x – 3)^{2} + y^{2} = 1\) , then a possible equation of L is

1. 1. \(x – \sqrt{3} y = 1\)
2. 2. \(x – \sqrt{3} y = 4\)
3. 3. \(x – \sqrt{3} y = -1\)
4. 4. \(x – \sqrt{3} y = 7\)

254. If the angle between the pair of tangents drawn to the circle \(x^{2} + y^{2} – 2x + 4y + 3 = 0\) from the point (6,-5) is \(\theta\) , then \(\cot \theta =\)

1. 1. \(\frac{8}{15}\)
2. 2. \(\frac{1}{4}\)
3. 3. 4
4. 4. \(\frac{15}{8}\)

255. The radius of a circle touching all the four circle \(\left(x\pm \lambda\right)^{2} + \left(y\pm \lambda\right)^{2} = \lambda^{2}\)

1. 1. \(2\sqrt{2}\lambda\)
2. 2. \((\sqrt{2} -1)\lambda\)
3. 3. \((2 + \sqrt{2})\lambda\)
4. 4. \((2 – \sqrt{2})\lambda\)

256. The equation of the circle inscribed in a square formed by the lines \(x + y – 2 = 0, x + y – 6 = 0\) , \(x – y + 1 = 0\) and \(x – y + 5 = 0\) is

1. 1. \(2x^{2} + 2y^{2} – 2x – 14y + 21 = 0\)
2. 2. \(x^{2} + y^{2} – x – 7y + 10 = 0\)
3. 3. \(2x^{2} + 2y^{2} – x – 7y + 21 = 0\)
4. 4. \(x^{2} + y^{2} – 2x – 14y + 10 = 0\)

257. Let the circle \(S \equiv x^{2} + y^{2} + 2gx + 2fy + c = 0\) touch the positive X- axis and the positive Y- axis. Let (2,4) be a point on the circle \(S = 0\) . If two such circles exist, then the difference of their areas is

1. 1. \(104\pi\)
2. 2. \(96\pi\)
3. 3. \(9\pi\)
4. 4. \(41\pi\)

258. If the equations \(2x – 3y + 3 = 0, 2x + y + 1 = 0\) and \(6x + 4y + 1 = 0\) represent the sides of a triangle, then the equation of the circle passing through the vertices of this triangle is

1. 1. \(4x^{2} + 4y^{2} + 9x – 10y + 7 = 0\)
2. 2. \(2x^{2} + 2y^{2} – 7x – 5y + 9 = 0\)
3. 3. \(8x^{2} + 8y^{2} + 18x – 20y + 17 = 0\)
4. 4. \(x^{2} + y^{2} + 3x – y + 13 = 0\)

259. If \(T_{1}T_{1}^{\prime}\) and \(T_{2}T_{2}^{\prime}\) are the common tangents of the circles \(S \equiv x^{2} + y^{2} – 2x – 4y – 4 = 0\) and \(S^{1} \equiv x^{2} + y^{2} + 4x + 4y + 4 = 0\) where \(T_{1}, T_{1}^{\prime}, T_{2}, T_{2}^{\prime}\) are the points of contact, then the distance between \(T_{1}\) and \(T_{1}^{\prime}\) is

1. 1. \(6\sqrt{6}\)
2. 2. \(5\sqrt{6}\)
3. 3. \(10\sqrt{6}\)
4. 4. \(2\sqrt{6}\)