Let a and b respectively be the semi-transverse and semi-conjugate axes of a standard hyperbola whose…

Given that focus of hyperbola is $S\left(5\mathrm{,\; 0}\right)$

$\Rightarrow ae=5$......$\left(\mathrm{i}\right)$

Therefore, Directrix $x=\frac{a}{\mathrm{e}}$

$\Rightarrow \frac{a}{e}=\frac{9}{5}$  ......$\left(\mathrm{ii}\right)$

From $\left(\mathrm{i}\right)$ and $\left(\mathrm{ii}\right)$, we get $\Rightarrow a^2=9$

$\Rightarrow a=3$ and $e=\frac{5}{3}$    

(It satisfies $9e^2-18e+5=0$ )

We also know that for an ellipse,

$b^2=a^2\left(e^2-1\right)$

$\Rightarrow b^2=16$

$\Rightarrow$ $a^2-b^2=9-16= -7$