The maximum area of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ which touches the line $y=3x+2$ is

$y=3x+2$ touches ellipse ; $m=3,c=2$

$c^2=a^2{m}^2+b^2\Rightarrow 4=9a^2+b^2$

$\begin{array}{l}AM\ge GM\Rightarrow \frac{9a^2+b^2}{2}\ge \sqrt{9a^2{b}^2}\Rightarrow 9a^2+b^2\ge 6ab\\ \Rightarrow 4\ge 6ab\\ \Rightarrow 2\ge 3ab\\ \Rightarrow \frac{2}{3}\ge ab\Rightarrow \frac{2\pi }{3}\ge \pi ab\end{array}$