A hyperbola passes through the foci of the ellipse x ^ 2 25 + y ^ 2...

For ellipse ${e}_{1} = \sqrt{1 - \dfrac{{b}^{2}}{{a}^{2}}} = \dfrac{3}{5}$

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

Let hyperbola be

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

$∵$ it passes through \left(3, 0\right) ⇒ \dfrac{9}{{a}^{2}} = 1$</p><p>$⇒ {a}^{2} = 9$</p><p>$⇒ {b}^{2} = {a}^{2}\left({e}^{2} - 1\right)

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

$∴$ Hyperbola is

$\dfrac{{x}^{2}}{9} - \dfrac{{y}^{2}}{16} = 1$

option \left(1\right)