Let H_1:x^2a^2-y^2b^2=1 and H_2:y^2B^2-x^2A^2=1 be two hyperbolas having lengths of latus …

Question

Let $H_1:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ and $H_2:\dfrac{y^2}{B^2}-\dfrac{x^2}{A^2}=1$ be two hyperbolas having lengths of latus recta $15\sqrt2$ and $12\sqrt5$ respectively. Let their eccentricities be $e_1=\sqrt{\dfrac52}$ 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 \_\_\_\_\_\_\_\_.

Answer: 55

Accepted Answer

$e_1^2=1+\dfrac{b^2}{a^2}=\dfrac52\Rightarrow \dfrac{b^2}{a^2}=\dfrac32$. With $\dfrac{2b^2}{a}=15\sqrt2$: solving gives $a=5\sqrt2,\ b^2=75$ (i.e. $b=5\sqrt3$).

Transverse axis of $H_1=2a=10\sqrt2$. Product of transverse axes $=100\sqrt{10}\Rightarrow 2B=10\sqrt5\Rightarrow B=5\sqrt5$.

Latus rectum of $H_2$: $\dfrac{2A^2}{B}=12\sqrt5 \Rightarrow A^2=6\sqrt5\cdot B=150$.

$$e_2^2=1+\frac{A^2}{B^2}=1+\frac{150}{125}=\frac{11}{5},\qquad 25e_2^2=55.$$

Solutions · 1