Question

Let $e_{1}$ and $e_{2}$ are eccentricities of $\frac{x^{2}}{18}+\frac{y^{2}}{4}=1$ and $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$ respectively and $\left(e_{1} , e_{2}\right)$ lies on $15x^{2}+3y^{2}=k$ . Find the value of $k$

Answer 1

For ellipse $4=18\left(1 - e_{1}^{2}\right)$
$e_{1}^{2}=\frac{7}{9}$
$e_{1}=\frac{\sqrt{7}}{3}$
for hyperbola :
$4=9\left(e_{2}^{2} - 1\right)$
$e_{2}=\frac{\sqrt{13}}{3}$
Put $\left(e_{1} , e_{2}\right)$ in
$15x^{2}+3y^{2}=k$
$15\times \frac{7}{9}+\frac{3 . 13}{9}=k$
$k=16$