CN-LEAGUE 2011 Problem 11

A line $\ell$ with slope of $\frac{1}{3}$ insects the ellipse $C:\frac{x^2}{36}+\frac{y^2}{4}=1$ at points $A,B$ and the point $P\left( 3\sqrt{2} , \sqrt{2}\right)$ is above the line $\ell$ .

[list](1) Prove that the locus of the incenter of triangle $PAB$ is a segment,(2) If $\angle APB=\frac{\pi}{3}$ , then find the area of triangle $PAB$ .[/list]