Equation of ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
$e=\frac{2 \sqrt{2}}{3}$
Length of major axis is the diameter of circle
$x^2+y^2=18$
$\therefore \quad 2 a=2(3 \sqrt{2})^2 \Rightarrow a=3 \sqrt{2}$
$e^2=1-\frac{b^2}{a^2}$
$\Rightarrow \quad \frac{8}{9}=1-\frac{b^2}{18} \Rightarrow b^2=2$
$\therefore$ Equation of ellipse is $\frac{x^2}{18}+\frac{y^2}{2}=1$
Equation of circle is $x^2+y^2=18$
Let $(h, k)$ be the pole, then equation of polar with respect to ellipse is
$\frac{x h}{18}+\frac{y k}{2}=1$
Since this is the tangent of circle
$\therefore \quad 3 \sqrt{2}=\left|\frac{1}{\sqrt{\frac{h^2}{(18)^2}+\frac{k^2}{4}}}\right|$
$\frac{h^2}{18^2}+\frac{k^2}{4}=\frac{1}{18}$
$\frac{h^2}{18}+\frac{9 k^2}{2}=1$
$\therefore$ Locus is $\frac{x^2}{18}+\frac{9 y^2}{2}=1$