A bisoptic (curve) of a hyperbola is a quartic called a spiric of Perseus (a kind of oval). It has generally 2 components: the internal one corresponds to obtuse angle alpha, the external one to the accute angle 180-alpha. In red on the worksheet. The green circle is teh director circle of teh given hyperbola (called also the orthoptic, i.e. the geometric locus of points through which pass pairs of perpendicular tangents.
Note that actually the isoptic curves are not the whole of the components: their points of intersection with the asymptotes do not belong to the isoptics, as through them passes a unique tangent to the hyperbola.
Reference:
Th. Dana-Picard, N. Zehavi and G. Mann (2014): Bisoptic curves of hyperbolas, International Journal of Mathematical Education in Science and Technology 45 (5), 762-781.