The director circle of $x^2$/$a^2$ + $y^2$/$b^2$ = 1 is $x^2$ + $y^2$ = $a^2$ + $b^2$. It is the locus of points from which the two tangents to the ellipse are perpendicular. This circle has the same centre as the ellipse and radius sqrt($a^2$ + $b^2$). The director circle exists only when $a^2$ + $b^2$ > 0 (always true for real ellipses).