Standard equation of hyperbola and mixed ellipse questionsHard

Hyperbola Latus Rectum and Eccentricity

Let ${H_1}:\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ and ${H_2}: - \frac{{{x^2}}}{{{A^2}}} + \frac{{{y^2}}}{{{B^2}}} = 1$ be two hyperbolas having length of latus rectums $15\sqrt 2$ and $12\sqrt 5$ respectively. Let their eccentricities be ${e_1} = \sqrt {\frac{5}{2}}$ and e₂ respectively. If the product of the lengths of their transverse axes is $100\sqrt {10}$ , then $25\,e_2^2$ is equal to ______

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Standard equation of hyperbola and mixed ellipse questionsHard

Hyperbola Latus Rectum and Eccentricity

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