A Circumcircle's Second Point

Let $A$ be the right vertex of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$), and let $PQ$ be a chord not through $A$ whose slope is a fixed constant $k$. Prove that the circumcircle of $\triangle APQ$ always passes through a second fixed point $B \neq A$, and find the coordinates of $B$.