Perpendicular Lines Through a Point

The ellipse $C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$) passes through $P(-2, 1)$ with eccentricity $\dfrac{\sqrt2}{2}$. Two perpendicular lines through $P$ meet $C$ again at $A$ and $B$. Prove that the line $AB$ passes through a fixed point, and find it.