Ellipse PYQs Set 1 (PDF)

Set 1 of 5 · Questions 1-30 of 146

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Ellipse · Set 1 (Q1-30)
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1. $The length of the latus rectum of an ellipse is 6 units and the distance between a focus and its nearest vertex on the major axis is \frac{5}{3} units. If e is the eccentricity of this ellipse, then e satisfies the equation$ (1) $25 x^{2} - 40 x + 16 = 0$ (2) $25 x^{2} + 40 x - 16 = 0$ (3) $25 x^{2} - 40 x - 16 = 0$ (4) $25 x^{2} + 40 x - 32 = 0$ 2. $Let x = 4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is \frac{1}{2} . If P (1, β), β > 0 is a point on this ellipse, then the equation of the normal to it at P is$ (1) $7 x - 4 y = 1$ (2) $4 x - 2 y = 1$ (3) $4 x - 3 y = 2$ (4) $8 x - 2 y = 5$ 3. $The equation \frac{x ^{2}}{2 - λ} - \frac{y ^{2}}{λ - 5} - 1 = 0 represents an ellipse if$ (1) $2 < λ < 5$ (2) $λ > 5$ (3) $2 > λ > 5$ (4) $λ < 2$ 4. $If the line x - 2 y = 12 is tangent to the ellipse \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 at the point (3, \frac{- 9}{2}) then the length of the latus rectum of the ellipse is:$ (1) $9$ (2) $122$ (3) $5$ (4) $83$ 5. $Let S and S^{’} be the foci of the ellipse \frac{x ^{2}}{25} + \frac{y ^{2}}{9} = 1 and P (α, β) be a point on the ellipse in the first quadrant. If (SP)^{2} + (S^{’} P)^{2} - SP \cdot S^{’} P = 37, then α^{2} + β^{2} is equal to :$ (1) $15$ (2) $11$ (3) $17$ (4) $13$ 6. $Foci of the ellipse 2 x^{2} + 3 y^{2} - 4 x - 12 y + 13 = 0 are$ (1) $(1 + \frac{1}{6}, 2) and (1 - \frac{1}{6}, 2)$ (2) $(\frac{1}{6} + 1, 2) and (\frac{1}{6} - 1, 2)$ (3) $(2, 1 + \frac{1}{6}) and (2, 1 - \frac{1}{6})$ (4) $(2, \frac{1}{6} + 1) and (2, \frac{1}{6} - 1)$ 7. $If the line αx + 4 y = 7, where α \in R, touches the ellipse 3 x^{2} + 4 y^{2} = 1 at the point P in the first quadrant, then one of the focal distances of P is :$ (1) $\frac{1}{3} + \frac{1}{2 7}$ (2) $\frac{1}{3} - \frac{1}{2 5}$ (3) $\frac{1}{3} + \frac{1}{2 5}$ (4) $\frac{1}{3} - \frac{1}{2 11}$ 8. $If the eccentricity and the length of the latus rectum of an ellipse \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 are \frac{3}{2} and 1 respectively, then the sum of the lengths of major axis and minor axis of the ellipse is$ (1) $6$ (2) $3$ (3) $10$ (4) $8$ 9. $The value of ' k' so that the line y = 2 x + k may touch the ellipse 3 x^{2} + 5 y^{2} = 15 is$ (1) $\pm 23$ (2) $\pm 13$ (3) $\pm 33$ (4) $\pm 32$ 10. $If the normal to the ellipse 3 x^{2} + 4 y^{2} = 12 at a point P on it is parallel to the line, 2 x + y = 4 and the tangent to the ellipse at P passes through Q (4, 4) then PQ is equal to:$ (1) $\frac{5 5}{2}$ (2) $\frac{61}{2}$ (3) $\frac{221}{2}$ (4) $\frac{157}{2}$ 11. $The sum of the squares of the eccentricities of the conics \frac{x ^{2}}{4} + \frac{y ^{2}}{3} = 1 and \frac{x ^{2}}{4} - \frac{y ^{2}}{3} = 1 is$ (1) $2$ (2) $\frac{7}{3}$ (3) $7$ (4) $3$ 12. $If the latus rectum of an ellipse is equal to half of minor axis, then its eccentricity is$ (1) $\frac{3}{4}$ (2) $\frac{3}{4}$ (3) $\frac{1}{4}$ (4) $\frac{3}{2}$ 13. $The radius of the circle passing through the foci of the ellipse \frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1 and having its centre at (0, 3) is$ (1) $6$ (2) $4$ (3) $3$ (4) $2$ 14. $The angle between the tangents drawn from the point (1, 2) to the ellipse 3 x^{2} + 2 y^{2} = 5 is$ (1) $tan^{- 1} (\frac{12 5}{5})$ (2) $tan^{- 1} (\frac{12 5}{13})$ (3) $\frac{π}{4}$ (4) $- \frac{π}{4}$ 15. $Let S (1, 0) and S^{’} (0, 1) be the foci of an ellipse such that SP + S^{’} P = 2 for any point P on the ellipse. If A (x_{1}, y_{1}) and A^{’} (x_{2}, y_{2}) are the end points of the major axis of this ellipse, then x_{1} + x_{2} =$ (1) $- 1/4$ (2) $-1$ (3) $1/4$ (4) $1$ 16. $If the co-ordinate of two points A and B are (7, 0) and (- 7, 0) respectively and P is any point on the conic, 9 x^{2} + 16 y^{2} = 144, then PA + PB is equal to:$ (1) $9$ (2) $16$ (3) $6$ (4) $8$ 17. $The equation of the ellipse with directrix 3 x + 4 y - 5 = 0, focus (1, 2) and eccentricity \frac{1}{2}, is$ (1) $x^{2} + 84 y^{2} - 24 xy - 360 y + 170 x + 475 = 0$ (2) $91 x^{2} + 84 y^{2} - 24 x y - 170 x - 360 y + 475 = 0$ (3) $91 x^{2} + 84 y^{2} - 24 xy - 170 x + 360 y + 475 = 0$ (4) $91 x^{2} + 84 y^{2} - 24 xy - 170 x - 360 y - 475 = 0$ 18. $If the tangent at a point P on the parabola y^{2} = 3 x is parallel to the line x + 2 y = 1 and the tangents at the points Q and R on the ellipse \frac{x ^{2}}{4} + \frac{y ^{2}}{1} = 1 are perpendicular to the line x - y = 2, then the area of the triangle PQR is$ (1) $\frac{9}{5}$ (2) $53$ (3) $\frac{3}{2} 5$ (4) $35$ 19. $The foci of the ellipse \frac{x ^{2}}{16} + \frac{y ^{2}}{b ^{2}} = 1 and the hyperbola \frac{x ^{2}}{144} - \frac{y ^{2}}{81} = \frac{1}{25} coincide. Then, the value of b^{2} is$ (1) $5$ (2) $7$ (3) $9$ (4) $1$ 20. $Let the length of the latus rectum of an ellipse with its major axis along x -axis and centre at the origin, be 8 . If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?$ (1) $(42, 22)$ (2) $(43, 22)$ (3) $(43, 23)$ (4) $(42, 23)$ 21. $The eccentricity of the ellipse \frac{x ^{2}}{36} + \frac{y ^{2}}{16} = 1 is$ (1) $\frac{2 5}{6}$ (2) $\frac{2 13}{4}$ (3) $\frac{2 5}{4}$ (4) $\frac{2 13}{6}$ 22. $P (θ_{1}) and Q (θ_{2}) are two points on the ellipse \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 with eccentricity e . If PSQ is a focal chord and tan (\frac{θ _{1}}{2}) tan (\frac{θ _{2}}{2}) = - (22 + 3), then e and S are$ (1) $\frac{1}{3}, (\frac{a}{3}, 0)$ (2) $\frac{1}{3}, (\frac{- a}{3}, 0)$ (3) $\frac{1}{2}, (\frac{a}{2}, 0)$ (4) $\frac{1}{2}, (\frac{- a}{2}, 0)$ 23. $Let the ellipse E : \frac{x ^{2}}{144} + \frac{y ^{2}}{169} = 1 and the hyperbola H : \frac{x ^{2}}{16} - \frac{y ^{2}}{λ ^{2}} = - 1 have the same foci. If e and L respectively denote the eccentricity and the length of the latus rectum of H , then the value of 24 (e + L) is :$ (1) $148$ (2) $126$ (3) $67$ (4) $296$ 24. $The length of the latus-rectum of the ellipse, whose foci are (2, 5) and (2, - 3) and eccentricity is \frac{4}{5}, is$ (1) $\frac{50}{3}$ (2) $\frac{18}{5}$ (3) $\frac{6}{5}$ (4) $\frac{10}{3}$ 25. $The distance between the focii of the ellipse x = 3 cos θ, y = 4 sin θ is$ (1) $27$ (2) $72$ (3) $7$ (4) $37$ 26. $Let the length of a latus rectum of an ellipse \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 be 10. If its eccentricity is the minimum value of the function f (t) = t^{2} + t + \frac{11}{12}, t \in R, then a^{2} + b^{2} is equal to:$ (1) $115$ (2) $120$ (3) $125$ (4) $126$ 27. $The eccentric angle of a point on the ellipse x^{2} + 3 y^{2} = 6 lying at a distance of 2 units from its centre is$ (1) $\frac{π}{6}$ (2) $\frac{π}{4}$ (3) $\frac{π}{3}$ (4) $\frac{π}{2}$ 28. $Find the eccentricity of an ellipse, if the length of its latus rectum is 4 units and distance between its vertex and the nearest focus is 3/2 units.$ (1) $\frac{1}{3}$ (2) $\frac{2}{3}$ (3) $\frac{1}{9}$ (4) $\frac{3}{4}$ 29. $IF S is the focus of the ellipse \frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1 lying on the positive X -axis P (θ) is a point on the ellipse such that SP = 1, then cos θ =$ (1) $\frac{1}{5}$ (2) $\frac{2}{5}$ (3) $\frac{1}{2}$ (4) $\frac{1}{3}$ 30. $The line x = 8 is the directrix of the ellipse E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 with the corresponding focus (2, 0) . If the tangent to E at the point P in the first quadrant passes through the point (0, 43) and intersects the x-axis at Q, then (3 PQ)^{2} is equal to$