9. (16 points) Let the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}-c^{2}}=1(a>$ $c>0)$ have its right focus at $F$. Draw a tangent line through a moving point $P$ on the ellipse, which intersects the $x$-axis at point $Q$. Find the equation of the locus of the centroid of $\triangle P Q F$.

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9. Let $P(m, n), F(c, 0), \triangle P Q F$ have centroid $G(x, y)$.
Then $l_{P Q}: \frac{m x}{a^{2}}+\frac{n y}{a^{2}-c^{2}}=1$.
Thus, $Q\left(\frac{a^{2}}{m}, 0\right)$.
Hence, $G\left(\frac{1}{3}\left(m+c+\frac{a^{2}}{m}\right), \frac{1}{3} n\right)$, eliminating $m$ and $n$ yields
$$
\frac{9 y^{2}}{a^{2}-c^{2}}+\frac{\left((3 x-c) \pm \sqrt{(c-3 x)^{2}-4 a^{2}}\right)^{2}}{4 a^{2}}=1 .
$$