Opposite Slopes Through a Focus

The ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{3} = 1$ has foci $F_1$ (left) and $F_2$ (right). A line $l$ with nonzero slope meets the ellipse at $A$ and $B$ ($A$ to the left of $B$), and the slopes of $F_1A$ and $F_1B$ are negatives of each other.

1. Prove that the line $l$ passes through a fixed point. 2. Let $M$ be the intersection of lines $F_1B$ and $F_2A$. Prove that $|MF_2| - |MF_1|$ is constant.