The director circle of $x^2$/$a^2$ - $y^2$/$b^2$ = 1 is $x^2$ + $y^2$ = $a^2$ - $b^2$. It exists only when a > b (i.e., e < sqrt(2)). When a = b (rectangular hyperbola, e = sqrt(2)), the director circle degenerates to the point (0,0) — only from the centre can perpendicular tangents be drawn. When a < b (e > sqrt(2)), no perpendicular tangent pair exists. This is a major difference from the ellipse, whose director circle always exists.