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.