A Parallel-Translate Fixed Point

An ellipse $E$ centered at the origin, with axes of symmetry the coordinate axes, passes through $A(0,-2)$ and $B\left(\dfrac 32,-1\right)$.

1. Find the equation of $E$. 2. A line through $P(1,-2)$ meets $E$ at two points $M$ and $N$. The horizontal line through $M$ meets the segment $AB$ at a point $T$, and $H$ satisfies $\overrightarrow{MT}=\overrightarrow{TH}$. Prove that the line $HN$ passes through a fixed point.