10. (20 points) Given that $AB$ is the major axis of the ellipse $\Gamma: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, and $CD$ is a chord of the ellipse $\Gamma$. The tangents at points $C$ and $D$ intersect at point $P$, the extension of $AD$ intersects the extension of $CB$ at point $E$, and the extension of $AC$ intersects the extension of $DB$ at point $F$. If $E$, $P$, and $F$ are collinear, find $\frac{EP}{PF}$.
10. As shown in Figure 3, let the center of the ellipse be $O$. Connect $O P$, intersecting $C D$ at point $M$.
Let $C\left(x_{1}, y_{1}\right), D\left(x_{2}, y_{2}\right), P\left(x_{0}, y_{0}\right)$.
Then the equation of line $C D$ is $\frac{x_{0} x}{a^{2}}+\frac{y_{0} y}{b^{2}}=1$.
From $\left\{\begin{array}{l}\frac{x_{0} x}{a^{2}}+\frac{y_{0} y}{b^{2}}=1, \\ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,\end{array}\right.$ we get
$$
\left(\frac{x_{0}^{2}}{a^{4}}+\frac{y_{0}^{2}}{a^{2} b^{2}}\right) x^{2}-\frac{2 x_{0}}{a^{2}} x+1-\frac{y_{0}^{2}}{b^{2}}=0 \text {. }
$$

By Vieta's formulas, the coordinates of the midpoint $N$ of $C D$ are
$$
x_{N}=\frac{x_{1}+x_{2}}{2}=\frac{x_{0}}{\frac{x_{0}^{2}}{a^{2}}+\frac{y_{0}^{2}}{b^{2}}}, y_{N}=\frac{y_{0}}{\frac{x_{0}^{2}}{a^{2}}+\frac{y_{0}^{2}}{b^{2}}} .
$$

Thus, $k_{O N}=\frac{y_{0}}{x_{0}}=k_{O P}$.
Therefore, points $O, N, P$ are collinear.
Hence, point $N$ coincides with $M$.
By Newton's theorem, $\frac{E P}{P F}=1$.