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

Answer: 16

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
$15x^2+3y^2=k$
$15\times \frac{7}{9}+\frac{3.13}{9}=k$
$k=16$