A hyperbola passes through the foci of the ellipse

For ellipse $e_{1}=\sqrt{1-\frac{b^{2}}{a^{2}}}=\frac{3}{5}$

for hyperbola $\mathrm{e}_{2}=\frac{5}{3}$

Let hyperbola be

$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$

$\because$ it passes through $(3,0) \Rightarrow \frac{9}{\mathrm{a}_{2}}=1$

$\Rightarrow \mathrm{a}^{2}=9$

$\Rightarrow \mathrm{b}^{2}=\mathrm{a}^{2}\left(\mathrm{e}^{2}-1\right)$

$=9\left(\frac{25}{9}-1\right)=16$

$\therefore$ Hyperbola is

$\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$ ... option 2 .