Opposite Slopes, Fixed Point

The ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$) has eccentricity $\dfrac{\sqrt2}{2}$; two non-vertex points $M, N$ on it satisfy $\angle MON = 90^\circ$ and

1. Find the ellipse. 2. A non-vertical line meets the ellipse at $P, Q$; $F$ is the right focus. If the slopes of $PF$ and $QF$ are negatives of each other, does $PQ$ pass through a fixed point? Find it or explain.