A Perpendicular-Bisector Fixed Point

The ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) passes through $(\sqrt{6}, 0)$ and has eccentricity $\dfrac{\sqrt{2}}{2}$. A line meets $E$ at points $A$ and $B$; let $l$ be the perpendicular bisector of segment $AB$, and let $M(x_0, y_0)$ be the intersection of line $AB$ with $l$.

1. Find the equation of the ellipse $E$. 2. If $x_0 = 1$, does the line $l$ pass through a fixed point? If so, find it; if not, explain why.