9. (16 points) Given $\odot O: x^{2}+y^{2}=r^{2}$, and ellipse $C$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, where $0<b<a<r$. From any point on $\odot O$, draw two tangent lines to ellipse $C$, and the line segment connecting the two points of tangency is called the corresponding chord of tangency. Try to find the region inside ellipse $C$ that never intersects with the above chord of tangency.
9. Let any point on $\odot O$ be $A(r \cos \alpha, r \sin \alpha)(\alpha \in[0,2 \pi))$.
Then the equation of the chord of the tangent points corresponding to $A$ on the ellipse is
$$
\frac{x r \cos \alpha}{a^{2}}+\frac{y r \sin \alpha}{b^{2}}=1 \text {. }
$$

Consider an ellipse $\frac{x^{2}}{c^{2}}+\frac{y^{2}}{d^{2}}=1$ (where $c, d$ are to be determined).
Thus, the tangent line at point $B(c \cos \alpha, d \sin \alpha)$ on this ellipse is
$$
\frac{x \cos \alpha}{c^{2}}+\frac{y d \sin \alpha}{d^{2}}=1 .
$$

By comparing equation (2) with equation (1), we take $c=\frac{a^{2}}{r}, d=\frac{b^{2}}{r}$. Then the chord of the tangent points for all such ellipses $C$ are
$$
\frac{r^{2} x^{2}}{a^{4}}+\frac{r^{2} y^{2}}{b^{4}}=1
$$

the tangents of which.
Therefore, equation (3) is the boundary of the desired region.
Thus, the desired region is the interior of the ellipse (3).