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).