Given equation of hyperbola and line are $\frac{x^{2}}{9}-\frac{y^{2}}{25}=1$ and $y=x$ respectively.
For intersection point of both curve put $y=x$, we get
$$
\begin{array}{l}
\frac{x^{2}}{9}-\frac{x^{2}}{25}=1 \\
\Rightarrow \quad x^{2}=\frac{9 \times 25}{16}=\left(\frac{15}{4}\right)^{2} \\
\Rightarrow \quad x=\pm \frac{15}{4} \text { and } y=\pm \frac{15}{4}
\end{array}
$$
$\therefore$ Intersetion points $P\left(\frac{15}{4}, \frac{15}{4}\right)$
and $\quad Q\left(\frac{-15}{4}, \frac{-15}{4}\right)$
Since, $P Q$ is major axis, then its length
$$
=2 \sqrt{2} \cdot \frac{15}{4}=\frac{15}{\sqrt{2}}
$$
and length of minor axis is $\frac{5}{\sqrt{2}}$ (given)
i.e., Major axis, $2 a=\frac{15}{\sqrt{2}} \Rightarrow a=\frac{15}{2 \sqrt{2}}$
and minor axis, $2 b=\frac{5}{\sqrt{2}} \Rightarrow b=\frac{5}{2 \sqrt{2}}$
$\therefore$ Eccentricity of an ellipse
$$
=\sqrt{\frac{a^{2}-b^{2}}{a^{2}}}=\sqrt{1-\left(\frac{b}{a}\right)^{2}}
$$
$=\sqrt{1-\left(\frac{1}{3}\right)^{2}}=\sqrt{\frac{8}{9}}=\frac{2 \sqrt{2}}{3}$