10. (20 points) Given that curve $C_{1}$ is an ellipse centered at the origin $O$ with foci $F_{1}$ and $F_{2}$ on the left and right respectively, and curve $C_{2}$ is a parabola with vertex at $O$ and focus at $F_{2}$. Point $A$ is the intersection of curves $C_{1}$ and $C_{2}$, and $\angle A F_{2} F_{1}$ is an obtuse angle. Given $\left|A F_{1}\right|=\frac{7}{2},\left|A F_{2}\right|=\frac{5}{2}$, a line $l$ passing through point $F_{2}$ and not perpendicular to the $x$-axis intersects curves $C_{1}$ and $C_{2}$ at points $B, C, D, E$ in sequence, where points $B, E$ are on curve $C_{1}$ and points $C, D$ are on curve $C_{2}$. Let $G$ be the midpoint of $C D$ and $H$ be the midpoint of $B E$. Find the value of $\frac{|B E|\left|G F_{2}\right|}{|C D|\left|H F_{2}\right|}$.
10. Let the equation of the ellipse be $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. Then $2 a=\left|A F_{1}\right|+\left|A F_{2}\right|=\frac{7}{2}+\frac{5}{2}=6 \Rightarrow a=3$. Let point $A(x, y), F_{1}(-c, 0), F_{2}(c, 0)$.
Then $(x+c)^{2}+y^{2}=\left(\frac{7}{2}\right)^{2}$, $(x-c)^{2}+y^{2}=\left(\frac{5}{2}\right)^{2}$.
Subtracting the above two equations gives $x c=\frac{3}{2}$.
By the definition of the parabola, we know $\left|A F_{2}\right|=x+c=\frac{5}{2}$. Hence $c=1, x=\frac{3}{2}$ or $x=1, c=\frac{3}{2}$ (discard).
Thus, the equation of the ellipse is $\frac{x^{2}}{9}+\frac{y^{2}}{8}=1$, and the equation of the parabola is $y^{2}=4 x$.
Let points $B\left(x_{1}, y_{1}\right), E\left(x_{2}, y_{2}\right)$,
$$
C\left(x_{3}, y_{3}\right), D\left(x_{4}, y_{4}\right) \text {, }
$$

The line $l: y=k(x-1)$, substituting into $\frac{x^{2}}{9}+\frac{y^{2}}{8}=1$, we get
$$
\begin{array}{l}
\left(8+9 k^{2}\right) y^{2}+16 k y-64 k^{2}=0 \\
\Rightarrow y_{1}+y_{2}=-\frac{16 k}{8+9 k^{2}}, y_{1} y_{2}=-\frac{64 k^{2}}{8+9 k^{2}} .
\end{array}
$$

Similarly, substituting $y=k(x-1)$ into $y^{2}=4 x$, we get
$$
\begin{array}{l}
k y^{2}-4 y-4 k=0 \\
\Rightarrow y_{3}+y_{4}=\frac{4}{k}, y_{3} y_{4}=-4 . \\
\text { Hence } \frac{|B E|\left|G F_{2}\right|}{|C D|\left|H F_{2}\right|}=\frac{\left|y_{1}-y_{2}\right|}{\left|y_{3}-y_{4}\right|} \cdot \frac{\frac{1}{2}\left|y_{3}+y_{4}\right|}{\frac{1}{2}\left|y_{1}+y_{2}\right|} \\
=\sqrt{\frac{\left(y_{1}+y_{2}\right)^{2}-4 y_{1} y_{2}}{\left(y_{1}+y_{2}\right)^{2}} \cdot \frac{\left(y_{3}+y_{4}\right)^{2}}{\left(y_{3}+y_{4}\right)^{2}-4 y_{3} y_{4}}} \\
=3 .
\end{array}
$$