Parabola PYQs Set 2 (PDF)

Set 2 of 7 · Questions 31-60 of 203

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Parabola · Set 2 (Q31-60)
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31. $The sum of the minimum and maximum distance of the point (4, - 3) to the circle x^{2} + y^{2} + 4 x - 10 y - 7 = 0, is$ (1) $10$ (2) $12$ (3) $16$ (4) $20$ 32. $A circle passes through the points (1, 2), (3, 4) . If its centre lies on the line x - y + 3 = 0 then its radius is equal to$ (1) $4$ (2) $3$ (3) $1$ (4) $2$ 33. $The locus of a point which divides the line segment joining the point (0, -1) and a point on the parabola, x^{2} = 4 y, internally in the ratio 1 : 2, is:$ (1) $9 x^{2} = 3 y + 2$ (2) $9 x^{2} = 12 y + 8$ (3) $9 y^{2} = 12 x + 8$ (4) $9 y^{2} = 3 x + 2$ 34. $If P (\frac{1}{2}, 4) and Q are the ends of a focal chord of the parabola y^{2} = 32 x and S is the focus of the parabola then SQ =$ (1) $\frac{17}{2}$ (2) $\frac{65}{2}$ (3) $136$ (4) $\frac{289}{2}$ 35. $The perimeter of a certain sector of a circle is equal to the length of the arc of the semicircle. Then the angle at the centre of the sector in radians is$ (1) $\frac{π}{3}$ (2) $\frac{2 π}{3}$ (3) $π - 2$ (4) $π + 2$ 36. $If x = 2 + 3 cos θ and y = 1 - 3 sin θ represent a circle then the centre and radius is$ (1) $(2, 1), 9$ (2) $(2, 1), 3$ (3) $(1, 2), \frac{1}{3}$ (4) $(- 2, - 1), 3$ 37. $If A = (0, - 2) and B is any point on the circle x^{2} + y^{2} - 2 x - 2 y + 1 = 0, then the maximum value of (AB)^{2} is$ (1) $51$ (2) $11 + 210$ (3) $9 + 35$ (4) $\frac{5 + 2 3}{2}$ 38. $Two circles of equal radius a cut orthogonally. If their centres are (2, 3) and (5, 6) then radical axis of these circles passes through the point$ (1) $(3 a, 5 a)$ (2) $(2 a, a)$ (3) $(a, \frac{5 a}{3})$ (4) $(a, a)$ 39. $If the straight line 3 x + 4 y = k touches the circle x^{2} + y^{2} = 16 x, then the values of k are$ (1) $16, - 64$ (2) $16 ,64$ (3) $- 16, - 64$ (4) $- 16, 64$ 40. $Let the point P of the focal chord PQ of the parabola y^{2} = 16 x be (1, - 4) . If the focus of the parabola divides the chord PQ in the ratio m : n, gcd (m, n) = 1, then m^{2} + n^{2} is equal to:$ (1) $17$ (2) $37$ (3) $10$ (4) $26$ 41. $If the point of intersection of the tangents drawn at the points where the line 5 x + y + 1 = 0 cuts the circle x^{2} + y^{2} - 2 x - 6 y - 8 = 0 is (a, b), then 5 a + b =$ (1) $3$ (2) $-44$ (3) $-1$ (4) $4$ 42. $The line through P (a, 2), where a \neq = 0, making an angle 45^{\circ} with the positive direction of the X -axis meets the curve \frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1 at A & D and the coordinate axes at B and C . If PA, PB, PC and PD are in a geometric progression, then 2 a =$ (1) $13$ (2) $7$ (3) $1$ (4) $-13$ 43. $If the circles x^{2} + y^{2} - 16 x - 20 y + 164 = r^{2} (r > 0) and x^{2} + y^{2} - 8 x - 14 y + 29 = 0 intersect in two distinct points, then the maximum possible integral value of r is$ (1) $1$ (2) $10$ (3) $-2$ (4) $2$ 44. $If L_{1}, L_{2} and L_{3} are the chords of contact of the three points (2, 0), (1, - 2) and (4, 4) respectively with respect to the circle x^{2} + y^{2} = 3, then L_{1}, L_{2}, L_{3} are$ (1) $concurrent lines$ (2) $sides of a right-angled triangle$ (3) $sides of an equilateral triangle$ (4) $parallel lines$ 45. $If the circle x^{2} + y^{2} - 4 x - 8 y - 5 = 0 intersects the line 3 x - 4 y - m = 0 in two distinct points, then the number of integral values of ' m' is$ (1) $52$ (2) $51$ (3) $50$ (4) $49$ 46. $If the circle x^{2} + y^{2} - 6 x - 12 y + 1 = 0 cuts another circle C orthogonally and the centre of the circle C is (- 4, 2), then its radius is$ (1) $21$ (2) $5$ (3) $\frac{3}{4}$ (4) $15$ 47. $The point of intersection of the direct common tangents drawn to the circles (x + 11)^{2} + (y - 2)^{2} = 225 and (x - 11)^{2} + (y + 2)^{2} = 25 is$ (1) $(\frac{- 11}{2}, 1)$ (2) $(- 22, 4)$ (3) $(\frac{11}{2}, - 1)$ (4) $(22, - 4)$ 48. $If the chord of contact of tangents from a point on the circle x^{2} + y^{2} = r_{1}^{2} to the circle x^{2} + y^{2} = r_{2}^{2} touches the circle x^{2} + y^{2} = r^{2} then r_{1}, r_{2} and r_{3} are in$ (1) $AP$ (2) $HP$ (3) $GP$ (4) $AGP$ 49. $If P(h, k) be point on the parabola x = 4 y^{2}, Which is nearest to the point Q (0, 33), then the distance of P from the directrix of the parabola y^{2} = 4 (x + y) is equal to$ (1) $2$ (2) $4$ (3) $8$ (4) $6$ 50. $If the three normal drawn to the parabola, y^{2} = 2 x pass through the point (a, 0) a \neq = 0, then 'a' must be greaterthan:$ (1) $\frac{1}{2}$ (2) $- 1 2$ (3) $- 1$ (4) $1$ 51. $If the parabolas y^{2} = 4 b (x - c) and y^{2} = 8 ax have a common normal, then which one of the following is a valid choice for the ordered triad (a, b, c) ?$ (1) $(\frac{1}{2}, 2, 3)$ (2) $(1, 1, 3)$ (3) $(\frac{1}{2}, 2, 0)$ (4) $(1, 1, 0)$ 52. $The distance of the point (6, - 22) from the common tangent y = mx + c, m > 0, of the curves x = 2 y^{2} and x = 1 + y^{2} is$ (1) $\frac{1}{3}$ (2) $5$ (3) $\frac{14}{3}$ (4) $53$ 53. $If the radical axis of the circles x^{2} + y^{2} + 2 gx + 2 fy + c = 0 and 2 x^{2} + 2 y^{2} + 3 x + 8 y + 2 c = 0 touches the circle x^{2} + y^{2} + 2 x + 2 y + 1 = 0, then (4 g - 3) (f - 2) =$ (1) $0$ (2) $-1$ (3) $1$ (4) $2$ 54. $Let P and Q be two external points of the circle S \equiv x^{2} + y^{2} - a^{2} = 0 . Let the chord of contact of the point P with respect to the circle S = 0 pass through Q . If l_{1} and l_{2} are the lengths of the tangents drawn from P and Q to the circle S = 0, then PQ =$ (1) $l_{1} + l_{2}$ (2) $\frac{l _{1} + l _{2}}{2}$ (3) $l_{1}^{2} + l_{2}^{2}$ (4) $l_{1}^{2} - 2 l_{1} + l_{2}^{2} - 2 l_{2}$ 55. $The locus of the point of intersection of the tangents drawn at the ends of a focal chord of the parabola x^{2} = - 8 y is$ (1) $x = - 2$ (2) $x = 2$ (3) $y = - 2$ (4) $y = 2$ 56. $Let A be the area of in-circle and A_{1}, A_{2}, A_{3} be the areas of ex-circles of a triangle. If A_{1} = 4, A_{2} = 9, A_{3} = 16, then A =$ (1) $81$ (2) $\frac{61}{169}$ (3) $\frac{144}{61}$ (4) $\frac{144}{169}$ 57. $The least and the greatest distances of the point (10, 7) from the circle x^{2} + y^{2} - 4 x - 2 y - 20 = 0 are$ (1) $5,15$ (2) $10,5$ (3) $15,20$ (4) $12,16$ 58. $The point (3, 4) is the focus and 2 x - 3 y + 5 = 0 is the directrix of a parabola. Its latusrectum is$ (1) $\frac{2}{13}$ (2) $\frac{4}{13}$ (3) $\frac{1}{13}$ (4) $\frac{3}{13}$ 59. $Let P (α, β) be a point on the parabola y^{2} = 4 x . If P also lies on the chord of the parabola x^{2} = 8 y whose mid point is (1, \frac{5}{4}) . Then (α - 28) (β - 8) is equal to______$ 60. $Let y^{2} = 12 x be the parabola and S be its focus. Let PQ be a focal chord of the parabola such that (SP) (SQ) = \frac{147}{4} . Let C be the circle described taking PQ as a diameter. If the equation of a circle C is 64 x^{2} + 64 y^{2} - αx - 643 y = β, then β - α is equal to .$