Mathematics - Hyperbola Question with Solution | TestHub

Let ${\mathrm{H}}_1:\frac{x^2}{{\mathrm{a}}^2}-\frac{y^2}{{\mathrm{b}}^2}=1$ and ${\mathrm{H}}_2:-\frac{x^2}{{\mathrm{A}}^2}+\frac{y^2}{{\mathrm{B}}^2}=1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their ecentricities be $e_1=\sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25{\mathrm{e}}_2^2$ is equal to ________.