The locus of the intersection of tangents which are at right angles is known as director circle of the hyperbola. The equation to the director circle is :
\(x^2+y^2\) = \(a^2-b^2\)
If \(b^2\) < \(a^2\), this circle is real ; If \(b^2\) = \(a^2\) the radius of the circle is zero & it reduces to a point circle at the origin. In this case the center is the only point from which the tangents at right angles can be drawn to the curve.
If \(b^2\) > \(a^2\), the radius of the circle is imaginary, so that there is no such circle & so no tangents at right angle can be drawn to the curve.