A Circle Inside an Ellipse with Perpendicular Radii

As shown in the figure, an ellipse is centered at the origin $O$ with its major axis on the $x$-axis and eccentricity $e = \dfrac{\sqrt{2}}{2}$. The vertical line through the left focus $F_1$ meets the ellipse at points $A$ and $A'$, with $|AA'| = 4$.

1. Find the standard equation of the ellipse. 2. A line perpendicular to the $x$-axis meets the ellipse at two distinct points $P$ and $P'$. A circle with center $Q$ passes through $P$ and $P'$ in such a way that every other point of the ellipse lies outside the circle. If $PQ \perp P'Q$, find the standard equation of the circle $Q$.