A Midpoint Locus and a Circle

A segment $AB$ of fixed length $4$ moves with its endpoints on the two intersecting lines $x + 2y = 0$ and $x - 2y = 0$ (one endpoint on each line). Let $G$ be the midpoint of $AB$.

1. Find the equation of the locus $C$ of $G$. 2. If the two tangent lines drawn from a point $P$ to the curve $C$ are perpendicular, prove that $P$ moves on a fixed circle.