An ellipse passes through the vertices of the hyperbola Let the major and minor axes of the ellipse coincide with the transverse and conjugate axes of the hyperbola . Let the product of the eccentricities of and be . If is the length of the latus re

Question

An ellipse $E : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, (a < b)$ passes through the vertices of the hyperbola $H : \frac{x^{2}}{49} - \frac{y^{2}}{64} = - 1 .$ Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$ . Let the product of the eccentricities of $E$ and $H$ be $\frac{1}{2}$ . If $l$ is the length of the latus rectum of the ellipse $E$ , then the value of $113 l$ is equal to

Smart Achievers · Accepted Answer

(2)

$Ellipse passes through (0, \pm 8)$

$b^{2} = 64 e_{E} = \sqrt{1 - \frac{b^{2}}{a^{2}}}, e_{H} = \frac{\sqrt{113}}{8}, λ = \frac{2 b^{2}}{a}$

Solution

1 1556

2 1552

3 1662

4 776

Ans.
(2)

$Ellipse passes through (0, \pm 8)$

$b^{2} = 64 e_{E} = \sqrt{1 - \frac{b^{2}}{a^{2}}}, e_{H} = \frac{\sqrt{113}}{8}, λ = \frac{2 b^{2}}{a}$

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