Ellipse PYQs Set 2 (PDF)

Set 2 of 5 · Questions 31-60 of 146

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Ellipse · Set 2 (Q31-60)
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31. $Let the line y = mx and the ellipse 2 x^{2} + y^{2} = 1 intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at (- \frac{1}{3 2}, 0) and (0, β) then β is equal to:$ (1) $\frac{2}{3}$ (2) $\frac{2 2}{3}$ (3) $\frac{2}{3}$ (4) $\frac{2}{3}$ 32. $The distance between the directrices of the ellipse \frac{x ^{2}}{36} + \frac{y ^{2}}{20} = 1 is$ (1) $9$ (2) $65$ (3) $18$ (4) $35$ 33. $A tangent is drawn at (33 cos θ, sin θ) (0 < θ < \frac{π}{2}) to the ellipse \frac{x ^{2}}{27} + \frac{y ^{2}}{1} = 1 . The value of θ for which the sum of the intercepts on the coordinate axes made by this tangent attains the minimum, is$ (1) $\frac{π}{6}$ (2) $\frac{π}{3}$ (3) $\frac{2 π}{3}$ (4) $\frac{2 π}{4}$ 34. $In an ellipse, two vertices are (5, 0) and (0, - 4) . Then the equation of the ellipse is$ (1) $\frac{x ^{2}}{16} + \frac{y ^{2}}{25} = 1$ (2) $\frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1$ (3) $\frac{x ^{2}}{5} + \frac{y ^{2}}{4} =$ (4) $x^{2} + y^{2} = 41$ 35. $The equations of the latusrectum of the ellipse 9 x^{2} + 25 y^{2} - 36 x + 50 y - 164 = 0 are$ (1) $x - 4 = 0, x + 2 = 0$ (2) $x - 6 = 0, x + 2 = 0$ (3) $x + 6 = 0, x - 2 = 0$ (4) $x + 4 = 0, x + 5 = 0$ 36. $If x cos α + y sin α = 4 is tangent to \frac{x ^{2}}{25} + \frac{y ^{2}}{9} = 1, then the value of α is$ (1) $tan^{- 1} (3/7)$ (2) $tan^{- 1} (3 /7)$ (3) $tan^{- 1} (7/3)$ (4) $tan^{- 1} (3/ 7)$ 37. $If an ellipse with foci at (3, 3) and (- 4, 4) is passing through the origin, then the eccentricity of that ellipse is$ (1) $\frac{5}{7}$ (2) $\frac{3}{7}$ (3) $\frac{1}{7}$ (4) $\frac{4}{7}$ 38. $Let the line 2 x + 3 y - k = 0, k > 0, intersect the x -axis and y -axis at the points A and B , respectively. If the equation of the circle having the line segment AB as a diameter is x^{2} + y^{2} - 3 x - 2 y = 0 and the length of the latus rectum of the ellipse x^{2} + 9 y^{2} = k^{2} is \frac{m}{n}, where m and n are coprime, then 2 m + n is equal to$ (1) $11$ (2) $10$ (3) $12$ (4) $13$ 39. $If m is the length of the latus rectum and n is the length of the major axis of the ellipse 25 x^{2} + 16 y^{2} - 150 x - 64 y - 111 = 0, then the ordered pair (m, n) =$ (1) $(\frac{16}{5}, 10)$ (2) $(\frac{32}{5}, 10)$ (3) $(\frac{25}{2}, 8)$ (4) $(\frac{25}{4}, 8)$ 40. $If a x^{2} + b y^{2} = 15 is the equation of the ellipse for which distance between its foci is 2 and distance between its directrices is 5 , then a + b =$ (1) $10$ (2) $8$ (3) $16$ (4) $12$ 41. $If the line 2 x - 3 y + 4 = 0 cuts the ellipse x = 3 cos θ, y = 5 sin θ in A and B and (α, β) is the midpoint of AB, then 3 β - 2 α =$ (1) $-4$ (2) $4$ (3) $-5$ (4) $5$ 42. $If a point P (x, y) moves along the ellipse \frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1 and if C is the center of the ellipse, then the sum of maximum and minimum values of CP is$ (1) $25$ (2) $9$ (3) $4$ (4) $5$ 43. $Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, \frac{x ^{2}}{4} + \frac{y ^{2}}{2} = 1 from any of its foci?$ (1) $(1, 2)$ (2) $(- 2, 3)$ (3) $(- 1, 3)$ (4) $(- 1, 2)$ 44. $The length of the chord of the ellipse \frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1, whose mid point is (1, \frac{2}{5}), is equal to :$ (1) $\frac{1691}{5}$ (2) $\frac{2009}{5}$ (3) $\frac{1741}{5}$ (4) $\frac{1541}{5}$ 45. $The locus of the point which moves such that the ratio of its distance from two fixed point in the plane is always a constant K (< 1) is$ (1) $hyperbola$ (2) $ellipse$ (3) $straight line$ (4) $circle.$ 46. $The length of the minor axis (along y -axis) of an ellipse in the standard form is \frac{4}{3} . If this ellipse touches the line, x + 6 y = 8; then its eccentricity is:$ (1) $\frac{1}{3} \frac{11}{3}$ (2) $\frac{5}{6}$ (3) $\frac{1}{2} \frac{11}{3}$ (4) $\frac{1}{2} \frac{5}{3}$ 47. $An ellipse is drawn by taking a diameter of the circle (x - 1)2 + y2 = 1 as its semi-minor axis and a diameter of the circle x2 + (y - 2)2 = 4 is semi-major axis. If the centre of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is :$ (1) $4x2 + y2 = 4$ (2) $x2 + 4y2 = 8$ (3) $4x2 + y2 = 8$ (4) $x2 + 4y2 = 16$ 48. $Let f (x) = x^{2} + 9, g (x) = \frac{x}{x - 9} and a = f \circ g (10), b = g \circ f (3) . If e and l denote the eccentricity and the length of the latus rectum of the ellipse \frac{x ^{2}}{a} + \frac{y ^{2}}{b} = 1, then 8 e^{2} + l^{2} is equal to.$ (1) $8$ (2) $16$ (3) $6$ (4) $12$ 49. $Let the ellipse E : x^{2} + 9 y^{2} = 9 intersect the positive x- and y-axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If the area of the triangle with vertices A, P and the origin O is \frac{m}{n}, where m and n are coprime, then m - n is equal to$ (1) $16$ (2) $15$ (3) $17$ (4) $18$ 50. $If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :$ (1) $\frac{5}{3}$ (2) $\frac{3}{2}$ (3) $\frac{1}{3}$ (4) $\frac{2}{5}$ 51. $The eccentricity of the ellipse 9 x^{2} + 25 y^{2} = 225 is$ (1) $\frac{4}{5}$ (2) $\frac{3}{4}$ (3) $\frac{3}{5}$ (4) $\frac{9}{16}$ 52. $If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is:$ (1) $\frac{3}{19}$ (2) $\frac{3}{16}$ (3) $\frac{4}{17}$ (4) $\frac{5}{7}$ 53. $The focal distances of the point (\frac{4}{5}, \frac{3}{5}) on the ellipse \frac{x ^{2}}{4} + \frac{y ^{2}}{9} = 1 are$ (1) $\frac{10}{3}, \frac{2}{3}$ (2) $3,1$ (3) $\frac{13}{3}, \frac{5}{3}$ (4) $4,2$ 54. $The parametric equations of the ellipse whose foci are (- 3, 0), (9, 0) and eccentricity is \frac{1}{3}, are$ (1) $x = 3 + 122 cos θ, y = 18 sin θ$ (2) $x = 3 + 18 cos θ, y = 122 sin θ$ (3) $x = 18 cos θ, y = 3 + 122 sin θ$ (4) $x = 3 + 42 cos θ, y = 18 sin θ$ 55. $The values of c such that the line y = 4 x + c touches the ellipse \frac{x ^{2}}{4} + \frac{v ^{2}}{1} = 1 is$ (1) $\pm 13$ (2) $\pm 7$ (3) $\pm 65$ (4) $\pm 7 4$ 56. $Statement I : The equation of the directrix of the ellipse 4 x^{2} + y^{2} - 8 x - 4 y + 4 = 0 is 3 y = 6 - 43 Statement II : The equation of the latus rectum of the ellipse x^{2} + 4 y^{2} - 4 x - 8 y + 4 = 0 is y = 2 + 3 Which of the above statement(s) is (are) true?$ (1) $Statement I is true, but Statement II is false$ (2) $Statement II is true, but Statement I is false$ (3) $Both Statement I and Statement II are true$ (4) $Both Statement I and Statement II are false$ 57. $The foci of the ellipse 9 x^{2} + 25 y^{2} = 225 are$ (1) $(\pm 4, 0)$ (2) $(\pm \frac{4}{5}, 0)$ (3) $(\pm \frac{12}{5}, 0)$ (4) $(\pm \frac{2}{5}, 0)$ 58. $If the tangents drawn from a point P to the ellipse 4 x^{2} + 9 y^{2} - 24 x + 36 y = 0 are perpendicular, then the locus of P is$ (1) $x^{2} + y^{2} - 6 x + 4 y + 13 = 0$ (2) $x^{2} + y^{2} - 6 x + 4 y - 13 = 0$ (3) $x^{2} + y^{2} = 26$ (4) $x^{2} + y^{2} + 6 x - 4 y - 13 = 0$ 59. $Let the product of the focal distances of the point (3, \frac{1}{2}) on the ellipse \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, (a > b), be \frac{7}{4} . Then the absolute difference of the eccentricities of two such ellipses is$ (1) $\frac{1 - 3}{2}$ (2) $\frac{3 - 2 2}{2 3}$ (3) $\frac{3 - 2 2}{3 2}$ (4) $\frac{1 - 2 2}{3}$ 60. $Let (h, k) lie on the circle C : x^{2} + y^{2} = 4 and the point (2 h + 1, 3 k + 2) lie on an ellipse with eccentricity e . Then the value of \frac{5}{e ^{2}} is equal to____.$