For the hyperbola, $\frac{x^2}{16}-\frac{y^2}{25}=1$, director circle is $x^2+y^2=-9$, which is not possible. Does that mean director circle is an optional feature of the hyperbola?
Edit: Director circle is the locus of point of intersection of perpendicular tangents. Its equation for the hyperbola is $x^2+y^2=a^2-b^2$.