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 $15\sqrt2$ and $12\sqrt5$ respectively. Let their eccentricities be $e_1=\sqrt{\frac52}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to ________.

Question

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] $e_1^2=1+\frac{b^2}{a^2}=\frac52\Rightarrow b^2=\frac32a^2$. Latus rectum of $H_1$: $\frac{2b^2}a=3a=15\sqrt2\Rightarrow a=5\sqrt2$. Transverse axes: $2a\cdot2B=100\sqrt{10}\Rightarrow B=\frac{25\sqrt{10}}{5\sqrt2}=5\sqrt5$. Latus rectum of $H_2$: $\frac{2A^2}B=12\sqrt5\Rightarrow A^2=150$. $e_2^2=1+\frac{A^2}{B^2}=1+\frac{150}{125}=\frac{11}5$, so $25e_2^2=55$. Answer: 55.

Explanation

Reviewed by Dr. Abubacker

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