Let the eccentricity of an ellipse is reciprocal to that of the hyperbola . If the ellipse intersects the hyperbola at right angles, then square of length of latus-rectum of the ellipse is _____.

2 Explanation: \begin{aligned} & \mathrm{E:}{\frac{x^{2}}{a^{2}}}+{\frac{y^{2}}{b^{2}}}=1 \\ & \mathrm{H}=x^{2}-y^{2}=\frac{1}{2}\Rightarrow e^{\prime}=\sqrt{2} \\ & e={\frac{1}{\sqrt{2}}}\Rightarrow e^{2}={\frac{1}{2}} \end{aligned} E and H are right angle they are confocal focus of hyperbola=Focus of ellipse

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let the eccentricity of an ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ is reciprocal to that of the hyperbola $2 x^{2} - 2 y^{2} = 1$ . If the ellipse intersects the hyperbola at right angles, then square of length of latus-rectum of the ellipse is _____.

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