JEE Mathematics Ellipse and Hyperbola MCQs Set 01

Choose the most appropriate option (a, b, c or d)

Question. The equation \( 2x^2 – 3xy + 5y^2 + 6x – 3y + 5 = 0 \) represents
(a) a parabola
(b) an ellipse
(c) a hyperbola
(d) a pair of straight lines
Answer: (b) an ellipse

Question. The set of real values of k for which the equation \( (k + 1)x^2 + 2(k – 1)xy + y^2 – x + 2y + 3 = 0 \) represents an ellipse is
(a) (0, 3)
(b) \( (-\infty, 0) \)
(c) \( (3, +\infty) \)
(d) \( (-\infty, \infty) \)
Answer: (a) (0, 3)

Question. The centre of the conic section \( 14x^2 – 4xy + 11y^2 – 44x – 58y + 71 = 0 \) is
(a) (2, 3)
(b) (2, -3)
(c) (-2, 3)
(d) (-2, -3)
Answer: (a) (2, 3)

Question. The eccentricity of the ellipse \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \) is
(a) \( \frac{\sqrt{5}}{2} \)
(b) \( \frac{2}{3} \)
(c) \( \frac{\sqrt{5}}{3} \)
(d) \( \frac{4}{9} \)
Answer: (c) \( \frac{\sqrt{5}}{3} \)

Question. The eccentricity of the hyperbola \( x^2 – 4y^2 = 16 \) is
(a) 2
(b) \( \frac{\sqrt{5}}{2} \)
(c) 4
(d) \( \frac{\sqrt{3}}{2} \)
Answer: (b) \( \frac{\sqrt{5}}{2} \)

Question. The eccentricity of the conic section \( 4(x^2 – y^2) = 1 \) is
(a) \( \sqrt{2} \)
(b) 2
(c) 4
(d) \( \frac{1}{4} \)
Answer: (a) \( \sqrt{2} \)

Question. The latus rectum of the conic section \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) whose eccentricity = 3, is
(a) \( \frac{2a^2}{b} \)
(b) \( \frac{2b}{a^2} \)
(c) \( 2a(1 - e^2) \)
(d) \( 2b(1 - e^2) \)
Answer: (c) \( 2a(1 - e^2) \)

Question. The ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) passes through the point (-3, 1) and has the eccentricity \( \sqrt{\frac{2}{5}} \). Then the major axis of the ellipse has the length
(a) \( 4\sqrt{\frac{2}{5}} \)
(b) \( 8\sqrt{\frac{2}{3}} \)
(c) \( 4\sqrt{\frac{2}{3}} \)
(d) \( 8\sqrt{\frac{2}{5}} \)
Answer: (b) \( 8\sqrt{\frac{2}{3}} \)

Question. The hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) passes through the point (2, 3) and has the eccentricity 2. Then the transverse axis of the hyperbola has the length
(a) 1
(b) 3
(c) 2
(d) 4
Answer: (c) 2

Question. In the ellipse \( x^2 + 3y^2 = 9 \) the distance between the foci is
(a) \( \sqrt{6} \)
(b) 3
(c) \( \frac{2}{3}\sqrt{6} \)
(d) \( 2\sqrt{6} \)
Answer: (d) \( 2\sqrt{6} \)

Question. The minor axis of the ellipse \( 9x^2 + 5y^2 = 30y \) is
(a) 6
(b) \( 2\sqrt{5} \)
(c) \( \sqrt{6} \)
(d) \( \sqrt{5} \)
Answer: (b) \( 2\sqrt{5} \)

Question. The foci of the ellipse \( 25x^2 + 36y^2 = 225 \) are
(a) \( \left( \pm \frac{1}{2}\sqrt{11}, 0 \right) \)
(b) \( \left( \pm \frac{5}{2}, 0 \right) \)
(c) \( \left( 0, \pm \frac{1}{2}\sqrt{11} \right) \)
(d) \( \left( 0, \pm \frac{5}{2} \right) \)
Answer: (a) \( \left( \pm \frac{1}{2}\sqrt{11}, 0 \right) \)

Question. If the eccentricity of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is e then the eccentricity of the hyperbola \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \) is
(a) e
(b) \( \frac{e}{\sqrt{e^2 - 1}} \)
(c) \( e\sqrt{e^2 - 1} \)
(d) \( e^2 – e \)
Answer: (b) \( \frac{e}{\sqrt{e^2 - 1}} \)

Question. If in an ellipse the minor axis = the distance between the foci and its latus rectum = 10 then the equation of the ellipse in the standard form is
(a) \( \frac{x^2}{(10)^2} + \frac{y^2}{(5\sqrt{2})^2} = 1 \)
(b) \( \frac{x^2}{(5\sqrt{2})^2} + \frac{y^2}{(10)^2} = 1 \)
(c) \( \frac{x^2}{25} + \frac{y^2}{(5/\sqrt{2})^2} = 1 \)
(d) none of the options
Answer: (a) \( \frac{x^2}{(10)^2} + \frac{y^2}{(5\sqrt{2})^2} = 1 \)

Question. If in a hyperbola the eccentricity is \( \sqrt{3} \), and the distance between the foci is 9 then the equation of the hyperbola in the standard form is
(a) \( \frac{x^2}{\left(\frac{\sqrt{3}}{2}\right)^2} - \frac{y^2}{\left(\frac{\sqrt{3}}{\sqrt{2}}\right)^2} = 1 \)
(b) \( \frac{x^2}{\left(\frac{3\sqrt{3}}{2}\right)^2} - \frac{y^2}{\left(\frac{3\sqrt{3}}{\sqrt{2}}\right)^2} = 1 \)
(c) \( \frac{x^2}{\left(\frac{3\sqrt{3}}{2}\right)^2} - \frac{y^2}{\left(\frac{3\sqrt{2}}{2}\right)^2} = 1 \)
(d) none of the options
Answer: (b) \( \frac{x^2}{\left(\frac{3\sqrt{3}}{2}\right)^2} - \frac{y^2}{\left(\frac{3\sqrt{3}}{\sqrt{2}}\right)^2} = 1 \)

Question. If in an ellipse, a focus is (6, 7), the corresponding directrix is x + y + 2 = 0 and the eccentricity = \( \frac{1}{2} \) then the equation of the ellipse is
(a) \( 7x^2 + 2xy + 7y^2 – 44x – 108y + 684 = 0 \)
(b) \( 7x^2 – 2xy + 7y^2 – 52x – 116y + 676 = 0 \)
(c) \( 9x^2 – 2xy + 9y^2 – 44x – 108y + 684 = 0 \)
(d) none of the options
Answer: (b) \( 7x^2 – 2xy + 7y^2 – 52x – 116y + 676 = 0 \)

Question. If for a rectangular hyperbola a focus is (1, 2) and the corresponding directrix is x + y = 1 then the equation of the rectangular hyperbola is
(a) \( x^2 – y^2 = 2 \)
(b) \( xy – y + 2 = 0 \)
(c) \( xy + y – 2 = 0 \)
(d) none of the options
Answer: (c) \( xy + y – 2 = 0 \)

Question. If two foci of an ellipse be (-2, 0) and (2, 0) and its eccentricity is \( \frac{2}{3} \) then the ellipse has the equation
(a) \( 5x^2 + 9y^2 = 45 \)
(b) \( 9x^2 + 5y^2 = 45 \)
(c) \( 5x^2 + 9y^2 = 90 \)
(d) \( 9x^2 + 5y^2 = 90 \)
Answer: (a) \( 5x^2 + 9y^2 = 45 \)

Question. If for a conic section a focus is (-1, 1), eccentricity = 3 and the equation of the corresponding directrix is x – y + 3 = 0 then the equation of the conic section is
(a) \( 7x^2 – 18xy + 7y^2 + 50x – 50y + 77 = 0 \)
(b) \( 7x^2 + 18xy + 7y^2 = 1 \)
(c) \( 7x^2 + 18xy + 7y^2 – 50x + 50y + 77 = 0 \)
(d) none of the options
Answer: (a) \( 7x^2 – 18xy + 7y^2 + 50x – 50y + 77 = 0 \)

Question. An ellipse having foci at (3, 1) and (1, 1) passes through the point (1, 3). Its eccentricity is
(a) \( \sqrt{2} - 1 \)
(b) \( \sqrt{3} - 1 \)
(c) \( \frac{1}{2}(\sqrt{2} - 1) \)
(d) \( \frac{1}{2}(\sqrt{3} - 1) \)
Answer: (a) \( \sqrt{2} - 1 \)

Question. A point on the ellipse \( \frac{x^2}{6} + \frac{y^2}{2} = 1 \) at a distance 2 from the centre of the ellipse has the eccentric angle
(a) \( \frac{\pi}{4} \)
(b) \( \frac{\pi}{3} \)
(c) \( \frac{\pi}{6} \)
(d) \( \frac{\pi}{2} \)
Answer: (a) \( \frac{\pi}{4} \)

Question. A point P on the ellipse \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \) has the eccentric angle \( \frac{\pi}{8} \). The sum of the distance of P from the two foci is
(a) 5
(b) 6
(c) 10
(d) 3
Answer: (c) 10

Question. If any point on a hyperbola has the coordinates \( (5\tan \phi, 4\sec \phi) \) then the eccentricity of the hyperbola is
(a) \( \frac{5}{4} \)
(b) \( \frac{\sqrt{41}}{5} \)
(c) \( \frac{25}{16} \)
(d) \( \frac{\sqrt{41}}{4} \)
Answer: (d) \( \frac{\sqrt{41}}{4} \)

Question. The slope of the diameter of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), whose length is the GM of the major and minor axes, is
(a) \( \sqrt{\frac{a}{b}} \)
(b) \( \sqrt{ab} \)
(c) \( \sqrt{\frac{b}{a}} \)
(d) \( \frac{a}{b} \)
Answer: (c) \( \sqrt{\frac{b}{a}} \)

Question. PP’ is a diameter of the ellipse \( b^2x^2 + a^2y^2 = a^2b^2 \) such that \( PP'^2 \) is the AM of the squares of the major and minor axes. Then the slope of PP’ is
(a) \( \frac{b}{a} \)
(b) \( \frac{a}{b} \)
(c) \( \frac{\pi}{4} \)
(d) \( \frac{\pi}{3} \)
Answer: (a) \( \frac{b}{a} \)

Question. P is a variable point on the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 2 \) whose foci are \( F_1 \) and \( F_2 \). The maximum area (in unit²) of the \( \Delta P F_1 F_2 \) is
(a) \( 2b\sqrt{a^2 – b^2} \)
(b) \( \sqrt{2}b\sqrt{a^2 – b^2} \)
(c) \( b\sqrt{a^2 – b^2} \)
(d) \( 2a\sqrt{a^2 – b^2} \)
Answer: (a) \( 2b\sqrt{a^2 – b^2} \)

Question. Which of the following points is an exterior point of the ellipse \( 16x^2 + 9y^2 – 16x – 32 = 0 \)?
(a) \( (\frac{1}{2}, 2) \)
(b) \( (\frac{1}{4}, 1) \)
(c) (3, -2)
(d) none of the options
Answer: (c) (3, -2)

Question. For the hyperbola \( \frac{x^2}{\cos^2 \alpha} - \frac{y^2}{\sin^2 \alpha} = 1 \), which of the following remains constant when \( \alpha \) varies?
(a) abscissa of vertices
(b) abscissa of foci
(c) eccentricity
(d) directrix
Answer: (b) abscissa of foci

Question. The foci of the ellipse \( \frac{x^2}{16} + \frac{y^2}{b^2} = 1 \) and the hyperbola \( \frac{x^2}{144/25} - \frac{y^2}{81/25} = 1 \) coincide. Then the value of \( b^2 \) is
(a) 5
(b) 7
(c) 9
(d) 1
Answer: (b) 7

Question. The equation of the tangent to the ellipse \( 4x^2 + 3y^2 = 12 \) at the point whose eccentric angle is \( \frac{\pi}{4} \) is
(a) \( \sqrt{3}x + 2y = 2\sqrt{6} \)
(b) \( 2x + \sqrt{3}y = 2\sqrt{6} \)
(c) \( 2x – \sqrt{3}y = 2\sqrt{6} \)
(d) none of the options
Answer: (b) \( 2x + \sqrt{3}y = 2\sqrt{6} \)