Parabola PYQs Set 4 (PDF)

Set 4 of 7 · Questions 91-120 of 203

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Parabola · Set 4 (Q91-120)
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91. $The distance of the point (6, 43) from the focus of y^{2} = 8 x is$ (1) $64$ (2) $4$ (3) $8$ (4) $2$ 92. $Let A be a point on the x -axi. common tangents are drawn from A to the curves x^{2} + y^{2} = 8 and y^{2} = 16 x . If one of these tangents touches the two curves at Q and R, then (QR)^{2} is equal to$ (1) $64$ (2) $76$ (3) $81$ (4) $72$ 93. $If the lengths of the tangents drawn from P to the circles x^{2} + y^{2} - 2 x + 4 y - 20 = 0 and x^{2} + y^{2} - 2 x - 8 y + 1 = 0 are in the ratio 2 : 1, then the locus P is$ (1) $x^{2} + y^{2} + 2 x + 12 y + 8 = 0$ (2) $x^{2} + y^{2} - 2 x + 12 y + 8 = 0$ (3) $x^{2} + y^{2} + 2 x - 12 y + 8 = 0$ (4) $x^{2} + y^{2} - 2 x - 12 y + 8 = 0$ 94. $The parabola with focus at (4, - 3) and vertex at (4, - 1) is$ (1) $x^{2} + 8 x + 6 y + 22 = 0$ (2) $x^{2} - 8 x - 10 y + 6 = 0$ (3) $x^{2} - 8 x - 16 y = 0$ (4) $x^{2} - 8 x + 8 y + 24 = 0$ 95. $If 3 x + y + k = 0 is a tangent to the circle x^{2} + y^{2} = 10, the values of k are,$ (1) $\pm 7$ (2) $\pm 5$ (3) $\pm 10$ (4) $\pm 9$ 96. $The tangents drawn at the extremeties of a focal chord of the parabola y^{2} = 16 x$ (1) $intersect at an angle of 60^{\circ}$ (2) $intersect at an angle of 45^{\circ}$ (3) $intersect on x = 0$ (4) $intersect on the line x + 4 = 0$ 97. $Let S = ((x, y) \in R^{2} : \frac{y ^{2}}{1 + r} - \frac{x ^{2}}{1 - r} = 1), where r \neq = \pm 1 Then S represents :$ (1) $a hyperbola whose eccentricity is \frac{2}{1 - r} when 0<r<1$ (2) $an ellipse whose eccentricity is \frac{2}{r + 1}, when r>1$ (3) $a hyperbola whose eccentricity is \frac{2}{r + 1}, when 0<r<1$ (4) $an ellipse whose eccentricity is \frac{1}{r + 1}, when r>1$ 98. $The circle S = 0 cuts the circle x^{2} + y^{2} - 4 x + 2 y - 7 = 0 orthogonally. If (2, 3) is the centre of the circle S = 0, then its radius is$ (1) $2$ (2) $1$ (3) $3$ (4) $4$ 99. $The total number of common tangents of x^{2} + y^{2} - 6 x - 8 y + 9 = 0 and x^{2} + y^{2} = 1 is$ (1) $4$ (2) $2$ (3) $3$ (4) $1$ 100. $If the length of the tangent from any point on the circle (x - 3)^{2} + (y + 2)^{2} = 5 r^{2} to the circle (x - 3)^{2} + (y + 2)^{2} = r^{2} is 16 units, then the area between the two circles in sq. units is$ (1) $32 π$ (2) $4 π$ (3) $8 π$ (4) $256 π$ 101. $The circles a x^{2} + a y^{2} + 2 g_{1} x + 2 f_{1} y + c_{1} = 0 and b x^{2} + b y^{2} + 2 g_{2} x + 2 f_{2} y + c_{2} = 0 (a \neq = 0 and b \neq = 0) cut orthogonally if$ (1) $g_{1} g_{2} + f_{1} f_{2} = a c_{1} + b c_{2}$ (2) $2 (g_{1} g_{2} + f_{1} f_{2}) = b c_{1} + a c_{2}$ (3) $b g_{1} g_{2} + a f_{1} f_{2} = b c_{1} + a c_{2}$ (4) $g_{1} g_{2} + f_{1} f_{2} = c_{1} + c_{2} .$ 102. $If the tangent at (1, 7) to the curve x^{2} = y - 6 touches the circle x^{2} + y^{2} + 16x + 12y + c = 0 then the value of c is :$ (1) $85$ (2) $95$ (3) $195$ (4) $185$ 103. $If (0, \frac{3}{4}) is the radical centre of the circles S \equiv x^{2} + y^{2} + αx + 6 y = 0, S^{’} \equiv x^{2} + y^{2} + 2 αx + αy + 6 = 0 and S^{’ ’} \equiv x^{2} + y^{2} + 6 αx - αy + 3 = 0 then the distance between the radical centre and the centre of the circle S^{’} = 0 is$ (1) $8$ (2) $15$ (3) $\frac{65}{4}$ (4) $\frac{5}{4}$ 104. $An equilateral triangle OAB is inscribed in the parabola y^{2} = 4 x with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is$ (1) $2 (8 - 33)$ (2) $2 (3 + 3)$ (3) $4 (6 + 3)$ (4) $4 (3 - 3)$ 105. $The number of possible common tangents that can be drawn to the circles x^{2} + y^{2} + 4 x - 6 y - 3 = 0 and x^{2} + y^{2} + 4 x - 2 y + 1 = 0 is$ (1) $4$ (2) $3$ (3) $1$ (4) $0$ 106. $Let x = 2 t, y = \frac{t ^{2}}{3} be a conic. Let S be the focus and B be the point on the axis of the conic such that S A ⊥ B A, where A is any point on the conic. If k is the ordinate of the centroid of △ S A B, then lim_{t \to 1} k is equal to$ (1) $\frac{17}{18}$ (2) $\frac{19}{18}$ (3) $\frac{11}{18}$ (4) $\frac{13}{18}$ 107. $If the equation of the parabola with vertex V (\frac{3}{2}, 3) and the directrix x + 2 y = 0 is α x^{2} β y^{2} - γxy - 30 x - 60 y + 225 = 0, then α + β + γ is equal to:$ (1) $7$ (2) $9$ (3) $8$ (4) $6$ 108. $Let the normal at the point P on the parabola y^{2} = 6x pass through the point (5, -8). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is :$ (1) $-3$ (2) $- \frac{9}{4}$ (3) $- \frac{5}{2}$ (4) $-2$ 109. $Let P be a point on the parabola y^{2} = 12 x and N be the foot of the perpendicular drawn from P on the axis of the parabola. A line is now drawn through the mid-point M of PN, parallel to its axis which meets the parabola at Q. If the y - intercept of the line NQ is \frac{4}{3}, then :$ (1) $MQ = \frac{1}{4}$ (2) $PN = 3$ (3) $PN = 4$ (4) $MQ = \frac{1}{3}$ 110. $The area of the circle passing through the points (5, \pm 2), (1, 2) is$ (1) $8 π$ (2) $4 π$ (3) $2 π$ (4) $16 π$ 111. $Tangent and normal are drawn at P(16,16) on the parabola y^{2} = 16x, which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and ∠CPB = θ , then a value of tan θ is :$ (1) $3$ (2) $\frac{4}{3}$ (3) $\frac{1}{2}$ (4) $2$ 112. $The equations of the sides AB and AC of a triangle ABC are (λ + 1) x + λy = 4 and λx + (1 - λ) y + λ = 0 respectively. Its vertex A is on the y-axis and its orthocentre is (1,2). The length of the tangent from the point C to the part of the parabola y^{2} = 6 x in the first quadrant is$ (1) $6$ (2) $22$ (3) $2$ (4) $4$ 113. $If the equation of the parabola, whose vertex is at (5, 4) and the directrix is 3x + y-29=0, is x^{2} + a y^{2} +bxy+cx+dy+k=0 then a + b + c+d + k is equal to$ (1) $575$ (2) $-575$ (3) $576$ (4) $-576$ 114. $An equilateral triangle is inscribed in the parabola y^{2} = 8 x, with one of its vertices is the vertex of the parabola. Then, length of the side of that triangle is$ (1) $243 units$ (2) $163 units$ (3) $83 units$ (4) $43 units$ 115. $Let the shortest distance from (a, 0), a > 0, to the parabola y^{2} = 4 x be 4. Then the equation of the circle passing through the point (a, 0) and the focus of the parabola, and having its centre on the axis of the parabola is:$ (1) $x^{2} + y^{2} - 10 x + 9 = 0$ (2) $x^{2} + y^{2} - 6 x + 5 = 0$ (3) $x^{2} + y^{2} - 4 x + 3 = 0$ (4) $x^{2} + y^{2} - 8 x + 7 = 0$ 116. $A line is a common tangent to the circle (x - 3)^{2} + y^{2} = 9 and the parabola y^{2} = 4 x If the two points ofcontact (a, b) and (c, d) are distinct and lie in the first quadrant, then 2 (a + c) is equal to$ 117. $Let the length of the focal chord PQ of the parabola y^{2} = 12 x be 15 units. If the distance of PQ from the origin is p , then 10 p^{2} is equal to _______$ 118. $Let r be the radius of the circle, which touches x -axis at point (a, 0), a < 0 and the parabola y^{2} = 9 x at the point (4, 6) . Then r is equal to________.$ 119. $Let P_{1} be a parabola with vertex (3, 2) and focus (4,4) and P_{2} be its mirror image with respect to the line x + 2y = 6. Then the directrix of P_{2} is x + 2y =_____.$ 120. $The ordinates of the points P and Q on the parabola with focus (3,0) and directrix x = -3 are in the ratio 3:1. If R =_{(} α, β_{)} is the point of intersection of the tangents to the parabola at P and Q, then \frac{β ^{2}}{α} is equal to$