 Solution

For a fixed point $F$ not on a fixed line $D$, the ellipse with focus of a conic $F$, directrix of a conic $D$ and eccentricity $0<e<1$ is the locus of points $P$ satisfyingSet $c=\sqrt{a^2-b^2}$ and $e=c/a$. One matched focus of a conic and directrix of a conic pair isIndeed the squared distance condition isExpanding and cancelling the terms linear in $x$ giveswhich is exactly $x^2/a^2+y^2/b^2=1$. Both sides of the unsquared distance condition are nonnegative, so squaring introduces no extra points. Reflecting across the $y$-axis gives the other matched pair: $F=(-c,0)$ and $D:x=-a^2/c$, with the same eccentricity.