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An ellipse passes through the foci of the hyperbola, $9 x 2 − 4 y 2 = 36$ and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is $1 / 2.$ then which of the following points doesn’t lie on the ellipse?

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An Intiative by Sri Chaitanya

132, 6

1213, 32

(13, 0)

392, 3

answer is B.

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Detailed Solution

$Given equation of hyperbola is$ $9 x 2 − 4 y 2 = 36$

$\Rightarrow \frac{x^{2}}{4} - \frac{y^{2}}{9} = 1$

$e_{1} = \sqrt{\frac{4 + 9}{4}} = \frac{\sqrt{13}}{2}$

$Foci of hyperbola is (\pm \sqrt{13}, 0)$

${ee}^{'} = \frac{1}{2}$

$So, eccentricity of ellipse is \frac{1}{\sqrt{13}}$

$Equation of the ellipse is \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

$ellipse passes through focus (\pm \sqrt{13}, 0) of hyperbola$

$a^{2} = 13, b^{2} = 12$

$\Rightarrow \frac{x^{2}}{13} + \frac{y^{2}}{12} = 1$

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