Parabola PYQs Set 3 (PDF)

Set 3 of 7 · Questions 61-90 of 203

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Parabola · Set 3 (Q61-90)
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61. $The number of common tangents to the circles x^{2} + y^{2} = 4 and x^{2} + y^{2} - 6 x - 8 y - 24 = 0 is,$ (1) $3$ (2) $4$ (3) $2$ (4) $1$ 62. $Let LL^{’} be the latus rectum and PQ be the focal chord of the parabola y^{2} = 16 x . If P = (1, 4) and P, L lie in the same quadrant then LQ =$ (1) $5$ (2) $20$ (3) $245$ (4) $125$ 63. $Let L_{1} be a tangent to the parabola y^{2} = 4 (x + 1) and L_{2} be a tangent to the parabola y^{2} = 8 (x + 2) such that L_{1} and L_{2} intersect at right angles. Then L_{1} and L_{2} meet on the straight line:$ (1) $2 x + 1 = 0$ (2) $x + 3 = 0$ (3) $x + 2 y = 0$ (4) $x + 2 = 0$ 64. $Let Z be the point of intersection of the axis and the directrix of the parabola 4 x^{2} - 12 x + 4 y + 5 = 0 . If S is its focus, then the point which divides SZ in the ratio 2 : 1 is$ (1) $(\frac{3}{2}, \frac{13}{12})$ (2) $(1, \frac{13}{12})$ (3) $(\frac{3}{4}, \frac{13}{4})$ (4) $(\frac{3}{2}, \frac{13}{4})$ 65. $The area of the circle having its centre at (3, 4) and touching the line 5 x + 12 y - 11 = 0 is$ (1) $16 π sq. units$ (2) $4 π sq. units$ (3) $12 π sq. units$ (4) $25 π sq. units$ 66. $If the parabola x^{2} = 4 ay passes through the point (2, 1), then the length of the latus rectum is$ (1) $4$ (2) $2$ (3) $8$ (4) $1$ 67. $If the line joining the points A (α) and B (β) on the ellipse \frac{x ^{2}}{25} + \frac{y ^{2}}{9} = 1 is a focal chord, then one possible value of cot \frac{α}{2} \cdot cot \frac{β}{2} is$ (1) $-3$ (2) $3$ (3) $-9$ (4) $9$ 68. $The number of common tangents to the circles x^{2} + y^{2} - y = 0 and x^{2} + y^{2} + y = 0 is$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 69. $Let the slope of a diameter AC of a circle of radius 25 units be \frac{3}{4} . If (3, 2) is the centre of the circle, A = (x_{1}, y_{1}) and C = (x_{2}, y_{2}) then \frac{x _{1} x _{2}}{y _{1} y _{2}} =$ (1) $\frac{- 13}{23}$ (2) $\frac{13}{23}$ (3) $\frac{- 23}{13}$ (4) $\frac{23}{13}$ 70. $If θ is the angle between the circles x^{2} + y^{2} - 2 x - 4 y - 4 = 0 and x^{2} + y^{2} - 8 x - 12 y + 43 = 0 then \| 7 sec θ - 18 cos θ \| =$ (1) $11$ (2) $9$ (3) $0$ (4) $1$ 71. $If one end of a focal chord AB of the parabola y^{2} = 8 x is at A (\frac{1}{2}, - 2), then the equation of the tangent to it at B is :$ (1) $x - 2 y + 8 = 0$ (2) $x + 2 y + 8 = 0$ (3) $2 x - y - 24 = 0$ (4) $2 x + y - 24 = 0$ 72. $If one end of a focal chord of the parabola, y^{2} = 16 x is at (1, 4) then the length of this focal chord is:$ (1) $25$ (2) $22$ (3) $24$ (4) $20$ 73. $If the tangent at the point P on the circle x^{2} + y^{2} + 6 x + 6 y = 2 meets the straight line 5 x - 2 y + 6 = 0 at a point Q on the y-axis, then the length of PQ is$ (1) $5$ (2) $6$ (3) $4$ (4) $3$ 74. $If (2 t^{2}, 4 t) is a point on the parabola y^{2} = 8 x such that its focal distance is 3 , then t =$ (1) $\pm 1$ (2) $\pm \frac{1}{2}$ (3) $\pm \frac{1}{3}$ (4) $\pm \frac{1}{2}$ 75. $Let x^{2} + y^{2} + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x^{2} at (2, 4). Then A + C is equal to______.$ (1) $16$ (2) $88/5$ (3) $72$ (4) $-8$ 76. $The point which has the same power with respect to each of the circles x^{2} + y^{2} - 8 x + 40 = 0, x^{2} + y^{2} - 5 x + 16 = 0 and x^{2} + y^{2} - 8 x + 16 y + 160 = 0 is$ (1) $(- 8, \frac{- 15}{2})$ (2) $(8, \frac{- 15}{2})$ (3) $(8, \frac{15}{2})$ (4) $(- 8, \frac{15}{2})$ 77. $If the circles x^{2} + y^{2} = 9 and x^{2} + y^{2} + 2 ax + 2 y + 1 = 0 touch each other internally, then a =$ (1) $\frac{4}{3}$ (2) $\frac{- 4}{3}$ (3) $\pm \frac{4}{3}$ (4) $1$ 78. $The locus of centres of the circles, which cut the circles x^{2} + y^{2} + 4 x - 6 y + 9 and x^{2} + y^{2} - 5 x + 4 y + 2 = 0 orthogonally, is$ (1) $3 x + 4 y - 5 = 0$ (2) $9 x - 10 y + 7 = 0$ (3) $9 x + 10 y - 7 = 0$ (4) $9 x - 10 y + 11 = 0$ 79. $Let the image of parabola x^{2} = 4 y, in the line x - y = 1 be (y + a)^{2} = b (x - c), a, b, c \in N . Then a + b + c is equal to$ (1) $8$ (2) $6$ (3) $12$ (4) $4$ 80. $If (- 1, - 1) is the radical centre of the circles x^{2} + y^{2} + 2 gx - 4 y + 4 = 0, x^{2} + y^{2} + 6 x + 2 fy + 12 = 0 and x^{2} + y^{2} + 10 y + 20 = 0 then g - f =$ (1) $0$ (2) $-1$ (3) $1$ (4) $2$ 81. $If the circles (x - 2)^{2} + (y - 3)^{2} = 25 and 25 x^{2} + 25 y^{2} - 40 x - 70 y - 160 = 0 touch internally at (α, β), then α + β is equal to$ (1) $0$ (2) $-2$ (3) $-1$ (4) $1$ 82. $If the circles C_{1} : x^{2} + y^{2} + 2 x + 4 y - 20 = 0, C_{2} : x^{2} + y^{2} + 6 x - 8 y + 9 = 0 have n common tangents and the length of the tangent drawn from the centre of similitude to the circle C_{2} is l then \frac{l}{n ^{2}} =$ (1) $439$ (2) $39$ (3) $\frac{39}{4}$ (4) $239$ 83. $The radius of the circle passing through the points (- 1, 1), (2, - 1) and (1, 0) is$ (1) $5$ (2) $\frac{130}{2}$ (3) $6$ (4) $\frac{145}{2}$ 84. $If the shortest distance of the parabola y^{2} = 4 x from the centre of the circle x^{2} + y^{2} - 4 x - 16 y + 64 = 0 is d, then d^{2} is equal to :$ (1) $16$ (2) $24$ (3) $20$ (4) $36$ 85. $Let y^{2} = 12 x be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x -axis such that ∠ OPA = 90^{\circ} . Then the locus of the centroid of such triangles OPA is :$ (1) $y^{2} - 2 x + 8 = 0$ (2) $y^{2} - 9 x + 6 = 0$ (3) $y^{2} - 4 x + 8 = 0$ (4) $y^{2} - 6 x + 4 = 0$ 86. $If the chord of contact of tangents from a point A to a given circle passes through B, then the circle with AB as a diameter will............$ (1) $Touch the given circle internally$ (2) $Cut the given circle orthogonally$ (3) $Touch the given circle externally$ (4) $Neither intersect nor touch the given circle$ 87. $If the eccentricity of the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 is \frac{5}{4} and 2 x + 3 y - 6 = 0 is a focal chord of the hyperbola, then the length of transverse axis is equal to$ (1) $\frac{24}{5}$ (2) $\frac{5}{24}$ (3) $\frac{12}{5}$ (4) $\frac{6}{5}$ 88. $Tangents are drawn to the hyperbola x^{2} - 9 y^{2} = 9 from point (3, 2) . Then, the area of the triangle formed by the tangents and the chord of contact is sq units.$ (1) $10$ (2) $6$ (3) $12$ (4) $8$ 89. $Let a line perpendicular to the line 2 x - y = 10 touch the parabola y^{2} = 4 (x - 9) at the point P . The distance of the point P from the centre of the circle x^{2} + y^{2} - 14 x - 8 y + 56 = 0 is__________$ 90. $Two tangent lines 1, and 1_{2} are drawn from the point (2, 0) to the parabola 2 y^{2} = -x. If the lines l_{1} and 1_{2} are also tangent to the circle (x - 5)^{2} + y^{2} = r, then 17r is equal to$