A Centroid Area Invariant

The ellipse $M \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{3} = 1$ (with $a > \sqrt{3}$) has right focus $F$, and the line $y = \dfrac{3}{\sqrt{7}}$ meets $M$ at $P$ and $Q$ with $PF \perp QF$.

1. Find the equation of the ellipse $M$. 2. Let $O$ be the origin, and let $A, B, C$ be three distinct points of the ellipse with $O$ the centroid of $\triangle ABC$. Is the area of $\triangle ABC$ constant? If so, find it; if not, explain why.