Let ellipse $\frac{x^2}{25}+\frac{y^2}{p^2}=1$ and hyperbola $\frac{x^2}{9}-\frac{y^2}{q^2}=1$ are c

Question

Let ellipse $\frac{x^2}{25}+\frac{y^2}{p^2}=1$ and hyperbola $\frac{x^2}{9}-\frac{y^2}{q^2}=1$ are confocal and meet at points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and D . If foci of ellipse subtend $90^{\circ}$ at point $A$ and $e_1$ and $e_2$ are eccentricities of ellipse and hyperbola then value of $25 \mathrm{e}_1^2+9 \mathrm{e}_2^2$ is

Professor PC Thomas Classes · Answer

$$
\begin{aligned}
& A S_1+A S_2=2 a=10 \
& A S_2-A S_1=2 A=6 \
& A S_2=8 \& A_1=2 \& S_1 S_2=\sqrt{68}
\end{aligned}
$$for ellipse $2 a_1=2 \sqrt{17} \Rightarrow e_1=\frac{\sqrt{17}}{5}$
for hyperbola $2 A e_2=2 \sqrt{17} \Rightarrow e_2=\frac{\sqrt{17}}{3}$