Midpoints of Reflected Chords of a Parabola

As shown in the figure, let $C:\ y^2 = 2x$ be a parabola, and let $l$ be a line through the point $A(2,0)$ that intersects $C$ at two points. Suppose there exist two distinct points $M$ and $N$ on $C$ that are symmetric with respect to the line $l$, and let $T$ denote the midpoint of $MN$.

1. Find the equation of the locus of $T$. 2. Find the maximum possible area of triangle $AMT$.