Lines Meeting at a Vertex

Let $A$ and $B$ be the left and right vertices of the ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$). The minor axis of $C$ has the same length as its focal distance, and $C$ passes through the point $(-\sqrt{2}, 1)$.

1. Find the standard equation of $C$. 2. A line through the right focus $F$ meets $C$ at two points $M, N$ (distinct from $A, B$). Lines $AM$ and $AN$, extended, meet the line $l \colon x = 2\sqrt{2}$ at two distinct points $P, Q$. Prove that lines $MQ$ and $NP$ intersect at the point $B$.