Let e_1 be the eccentricity of the hyperbola [(x^2)/(16)-(y^2)/(9)=1] and e_2 be the eccen…

Question

Let $e_1$ be the eccentricity of the hyperbola $$\frac{x^2}{16}-\frac{y^2}{9}=1$$ and $e_2$ be the eccentricity of the ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \mathrm{a} > \mathrm{b}$$, which passes through the foci of the hyperbol...

Accepted Answer

$\begin{gathered} H:\frac{{x}^{2}}{16}-\frac{{y}^{2}}{9}=1\, {e}_{1}=\frac{5}{4} \ 	herefore {e}_{1}{e}_{2}=1\Rightarrow {e}_{2}=\frac{4}{5} \end{gathered}$

Also, ellipse is passing through $(\pm 5,0)$

$\begin{gathered} 	herefore a=5b=3 \ E:\frac{{x}^{2}}{25}+\frac{{y}^{2}}{9}=1 \end{gathered}$

End point of chord are $(\pm \frac{5\sqrt{5}}{3},2)$

$	herefore {L}_{PQ}=\frac{10\sqrt{5}}{3}$

Solutions · 1