Ellipse PYQs Set 3 (PDF)

Set 3 of 5 · Questions 61-90 of 146

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Ellipse · Set 3 (Q61-90)
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61. $Let the length of the latus rectum of an ellipse \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 be equal to the length of its semi-major axis. If the radius of its director circle is 3 and e is its eccentricity, then the length of its latus rectum is$ (1) $\frac{1}{a}$ (2) $\frac{1}{b}$ (3) $\frac{1}{e}$ (4) $\frac{1}{ab}$ 62. $A particle is travelling in clockwise direction on the ellipse \frac{x ^{2}}{100} + \frac{y ^{2}}{25} = 1 . If the particle leaves the ellipse at the point (- 8, 3) on it and travels along the tangents to the ellipse at that point then the point where the particle crosses the Y-axis is$ (1) $(0, \frac{7}{3})$ (2) $(0, \frac{25}{3})$ (3) $(0, 9)$ (4) $(0, \frac{- 25}{3})$ 63. $The number of values of ' c' such that the line y = 4 x + c touches the curve \frac{x ^{2}}{4} + y^{2} = 1 is$ (1) $infinite$ (2) $0$ (3) $1$ (4) $2$ 64. $If the sum of the distances from the foci to the centre O (0, 0) of an ellipse is 86 units and the area of the smallest rectangle in which that ellipse is inscribed is 80 sq. units, then the equation of such an ellipse is$ (1) $\frac{x ^{2}}{100} + \frac{y ^{2}}{64} = 1$ (2) $\frac{x ^{2}}{100} + \frac{y ^{2}}{16} = 1$ (3) $\frac{x ^{2}}{10} + \frac{y ^{2}}{4} = 1$ (4) $\frac{x ^{2}}{100} + \frac{y ^{2}}{4} = 1$ 65. $Let X - axis be the transverse axis and Y-axis be the conjugate axis of a hyperbola H . Let x^{2} + y^{2} = 16 be the director circle of H . If the perpendicular distance from the centre of H to its latus rectum is 34 then a + b =$ (1) $8$ (2) $9$ (3) $5$ (4) $7$ 66. $If the point of intersections of the ellipse \frac{x ^{2}}{16} + \frac{y ^{2}}{b ^{2}} = 1 and the circle x^{2} + y^{2} = 4 b, b > 4 lie on the curve y^{2} = 3 x^{2}, then b is equal to:$ (1) $12$ (2) $5$ (3) $6$ (4) $10$ 67. $If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity of the ellipse satisfies :$ (1) $e^{2} + 2 e - 1 = 0$ (2) $e^{2} + e - 1 = 0$ (3) $e^{4} + 2 e^{2} - 1 = 0$ (4) $e^{4} + e^{2} - 1 = 0$ 68. $The centre of the ellipse \frac{( x + y - 3 ) ^{2}}{9} + \frac{( x - y + 1 ) ^{2}}{16} = 1, is$ (1) $(- 1, 2)$ (2) $(1, - 2)$ (3) $(- 1, - 2)$ (4) $(1, 2)$ 69. $If the tangent drawn to the parabola y^{2} = 4 x at (t^{2}, 2 t) is the normal to the ellipse 4 x^{2} + 5 y^{2} = 20 at (5 cos θ, 2 sin θ), then$ (1) $5 t^{4} + 4 t^{2} = 1$ (2) $\frac{5}{t ^{4}} + \frac{100}{t ^{2}} = 1$ (3) $t = sin θ$ (4) $cos θ = t + 1$ 70. $Let L be a tangent line to the parabola y^{2} = 4 x - 20 at (6, 2) . If L is also a tangent to the ellipse \frac{x ^{2}}{2} + \frac{y ^{2}}{b} = 1, then the value of b is equal to:$ (1) $11$ (2) $14$ (3) $16$ (4) $20$ 71. $The equation of the chord, of the ellipse \frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1, whose mid-point is (3, 1) is:$ (1) $48 x + 25 y = 169$ (2) $5 x + 16 y = 31$ (3) $25 x + 101 y = 176$ (4) $4 x + 122 y = 134$ 72. $Let x^{2} + y^{2} = 20 be the director circle of an ellipse E whose major axis is X -axis and minor axis is Y -axis. If the length of the latus rectum of E is 2 , then the distance between its foci is$ (1) $45$ (2) $43$ (3) $42$ (4) $3$ 73. $The eccentric angle of the point (2, 3) lying on \frac{x ^{2}}{16} + \frac{y ^{2}}{4} = 1 is$ (1) $\frac{π}{2}$ (2) $\frac{π}{4}$ (3) $\frac{π}{6}$ (4) $\frac{π}{3}$ 74. $If the maximum distance of normal to the ellipse \frac{x ^{2}}{4} + \frac{y ^{2}}{b ^{2}} = 1, b < 2 from the origin is 1, then the eccentricity of the ellipse is$ (1) $\frac{1}{2}$ (2) $\frac{3}{2}$ (3) $\frac{1}{2}$ (4) $\frac{3}{4}$ 75. $In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at (0, 53) then the length of its latus rectum is:$ (1) $10$ (2) $5$ (3) $8$ (4) $6$ 76. $Assertion : The image of \frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1 in the line x+y=10 is \frac{( x - 10 ) ^{2}}{16} + \frac{( y - 10 ) ^{2}}{25} = 1 . Reason : The image of a curve ' C ' in a line L is the locus of the image of every point of C with respect to the line L.The correct option among the following is:$ (1) $(A) is true, (R) is true and (R) is the correct explanation for (A)$ (2) $(A) is true, (R) is true but (R) is not the correct explanation for (A)$ (3) $(A) is true but (R) is false$ (4) $(A) is false but (R) is true$ 77. $If 3 x + 4 y = 122 is a tangent to the ellipse \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{9} = 1 for some a \in R, then the distance between the foci of the ellipse is:$ (1) $25$ (2) $23$ (3) $27$ (4) $4$ 78. $The line 12x cos θ + 5y sin θ = 60 is tangent to which of the following curves?$ (1) $x^{2} + y^{2} = 169$ (2) $144 x^{2} + 25 y^{2} = 3600$ (3) $25 x^{2} + 12 y^{2} = 3600$ (4) $x^{2} + y^{2} = 60$ 79. $Let A be a vertex of the ellipse S = \frac{x ^{2}}{4} + \frac{y ^{2}}{9} - 1 = 0 and F be a focus of the ellipse S^{’} = \frac{x ^{2}}{9} + \frac{y ^{2}}{4} - 1 = 0 . Let P be a point on the major axis of the ellipse S^{’} = 0, which divides OF in the ratio 2 : 1 (O is the origin). If the length of the chord of the ellipse S = 0 through A and P is \frac{3 101}{k}, then k =$ (1) $5$ (2) $4$ (3) $7$ (4) $8$ 80. $The length of the chord of the ellipse \frac{x ^{2}}{4} + \frac{y ^{2}}{2} = 1, whose mid-point is (1, \frac{1}{2}), is:$ (1) $\frac{5}{3} 15$ (2) $\frac{1}{3} 15$ (3) $\frac{2}{3} 15$ (4) $15$ 81. $Let the length of the latus rectum of an ellipse \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, (a > b), be 30 . If its eccentricity is the maximum value of the function f (t) = - \frac{3}{4} + 2 t - t^{2}, then (a^{2} + b^{2}) is equal to$ (1) $276$ (2) $256$ (3) $516$ (4) $496$ 82. $If the tangents on the ellipse 4 x^{2} + y^{2} = 8 at the points (1, 2) and (a, b) are perpendicular to each other, then a^{2} is equal to:$ (1) $\frac{128}{17}$ (2) $\frac{64}{17}$ (3) $\frac{4}{17}$ (4) $\frac{2}{17}$ 83. $Let \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 (a > b) be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function, φ (t) = \frac{5}{12} + t - t^{2}, then a^{2} + b^{2} is equal to:$ (1) $135$ (2) $116$ (3) $126$ (4) $145$ 84. $The equation of the ellipse in the standard form whose length of the latus rectum is 4 and whose distance between the foci is 42, is$ (1) $\frac{x ^{2}}{2} + \frac{y ^{2}}{3} =$ (2) $2 x^{2} + y^{2} = 8$ (3) $x^{2} + 2 y^{2} = 16$ (4) $x^{2} + 5 y^{2} = 25$ 85. $If the area of the auxiliary circle of the ellipse \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 (a > b) is twice the area of the ellipse, then the eccentricity of the ellipse is$ (1) $\frac{1}{2}$ (2) $\frac{3}{2}$ (3) $\frac{1}{3}$ (4) $\frac{1}{2}$ 86. $If P (θ) and Q (\frac{π}{2} + θ) are two points on the ellipse \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 and the locus of midpoint of PQ is \frac{x ^{2}}{α ^{2}} + \frac{y ^{2}}{β ^{2}} = 1, then \frac{a + b}{α + β} =$ (1) $\frac{1}{2}$ (2) $3$ (3) $\frac{1}{3}$ (4) $2$ 87. $Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of P . If p + q = 126, then the eccentricity of the ellipse \frac{x ^{2}}{16} + \frac{y ^{2}}{n} = 1 is:$ (1) $\frac{1}{2}$ (2) $\frac{1}{2}$ (3) $\frac{7}{4}$ (4) $\frac{3}{4}$ 88. $If the tangent to the parabola y^{2} = x at a point (α, β), (β > 0) is also a tangent to the ellipse, x^{2} + 2 y^{2} = 1, then α is equalto:$ (1) $2 - 1$ (2) $22 - 1$ (3) $22 + 1$ (4) $2 + 1$ 89. $If the points of intersection of two distinct conics x^{2} + y^{2} = 4 b and \frac{x ^{2}}{16} + \frac{y ^{2}}{b ^{2}} = 1 lie on the curve y^{2} = 3 x^{2}, then 33 times the area of the rectangle formed by the intersection points is_________$ 90. $Let a tangent to the Curve 9 x^{2} + 16 y^{2} = 144 intersect the coordinate axes at the points A and B. Then, the minimum length of the line segment AB is$