10. Given $F_{1}$ and $F_{2}$ are the left and right foci, respectively, of

the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$,
the circle with diameter $F_{1} F_{2}$ intersects the hyperbola $C$ at point $P$ in the second quadrant. If the eccentricity of the hyperbola is 5, then the value of $\cos \angle P F_{2} F_{1}$ is ( ).
(A) $\frac{3}{5}$
(B) $\frac{3}{4}$
(C) $\frac{4}{5}$
(D) $\frac{5}{6}$
10. C.

Let $P F_{1}=x$.
Since point $P$ is on the hyperbola, we know $P F_{2}=2 a+x$. Since point $P$ is on the circle, we know $P F_{1}^{2}+P F_{2}^{2}=F_{1} F_{2}^{2}$.
Then $x=-a+\sqrt{b^{2}+c^{2}}$
$\Rightarrow P F_{1}=-a+\sqrt{b^{2}+c^{2}}$.
Also, $\frac{c}{a}=5$, so $\cos \angle P F_{2} F_{1}=\frac{4}{5}$.