9. $P P^{\prime}$ is any chord of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ parallel to the $y$-axis, $A, A^{\prime}$ are the two vertices on the $x$-axis, the lines $A^{\prime} P$ and $A P^{\prime}$ intersect at $Q$, find the locus of $Q$.
9. Let the coordinates of point $P$ be $(a \cos \alpha, b \sin \alpha)$ (parametric representation of an ellipse), then the coordinates of point $P^{\prime}$ are $(a \cos \alpha, -b \sin \alpha)$. The equation of line $A^{\prime} P$ is $y=\frac{b \sin \alpha}{a(1+\cos \alpha)}(x+a)$,
the equation of line $A P^{\prime}$ is $y=\frac{b \sin \alpha}{a(1-\cos \alpha)}(x-a)$. Multiplying the two equations gives $y^{2}=\frac{b^{2}}{a^{2}}\left(x^{2}-a^{2}\right)$, which simplifies to $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. Therefore, the locus of point $Q$ is the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$.