A Half-Area Chord

A line $l$ meets the ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ at points $P$ and $Q$, the triangle $POQ$ (with $O$ the origin) has area $\dfrac{ab}{2}$, and $M$ is the midpoint of segment $PQ$. Prove that the sum of the squares of the $x$-coordinates of $P, Q$ and the sum of the squares of their $y$-coordinates are both constants, and find the maximum value of $|OM|\cdot|PQ|$.