Perpendiculars from the Right Vertex

Let $A$ be the right vertex of the ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$). Two mutually perpendicular lines $AP$ and $AQ$ through $A$ meet the ellipse again at $P$ and $Q$.

1. Prove that the line $PQ$ passes through a fixed point $R$, and find its coordinates. 2. Find the maximum possible area of $\triangle APQ$.