A Ladder Point Locus

A segment $AB$ of length $3$ has $A$ on the $x$-axis and $B$ on the $y$-axis, and a point $P$ on the line $AB$ satisfies $\overrightarrow{BP} = 2\overrightarrow{PA}$.

1. Find the equation of the locus of $P$. 2. Call this locus the curve $C$. A line $l$ through $Q(2, 1)$ meets $C$ at $M$ and $N$; through $M$, the line $l'$ of slope $-\dfrac{1}{2}$ meets $C$ again at $R$. Prove that the intersection of line $NR$ with line $OQ$ is a fixed point (where $O$ is the origin), and find that point.