Tangents and a Focus

Let $F$ be the focus of the parabola $C \colon y = x^2$, and let $P$ move along the line $l \colon x - y - 2 = 0$. From $P$, the two tangent lines $PA, PB$ touch the parabola at $A$ and $B$.

1. Find the equation of the locus of the centroid $G$ of $\triangle APB$. 2. Prove that $\angle PFA = \angle PFB$.