Answer
The domain of the given function is
$$
\left\{(x, y) | \frac{1}{9} x^{2}+y^{2}<1\right\}.
$$
Work Step by Step
$$\ln (9-x^{2}-9y^{2})$$
This function is defined only when:
$$
9-x^{2}-9y^{2} \gt 0
$$
$\Rightarrow$
$$
\frac{1}{9}x^{2}+y^{2} \lt 1
$$
Therefore, the domain of the given function is
$$
\left\{(x, y) | \frac{1}{9} x^{2}+y^{2}<1\right\}.
$$
