An Isosceles with a Directrix

An ellipse $C$ is centered at the origin with foci on the $x$-axis; its right focus is $F$, the line $l \colon x = 4$ is its right directrix, and the distance from $F$ to $l$ is $3$.

1. Find the equation of the ellipse $C$. 2. Let $P$ move along the ellipse, and let $M$ be the foot of the perpendicular from $P$ to $l$. Is there a point $P$ making $\triangle FPM$ isosceles? If so, find all such $P$; if not, explain why.