Definition of the rectangular hyperbola

A notable special case occurs when the two semi-axes coincide.

Definition — Rectangular hyperbola

A hyperbola is rectangular if $a = b$. In this case the canonical equation becomes $x^2 - y^2 = a^2,$ and the asymptotes are the bisectors of the quadrants $y = \pm x$ (perpendicular to each other).

The perpendicularity of the asymptotes is the geometric feature that immediately distinguishes a rectangular hyperbola from any other.

Links

Topics: Homographic hyperbola
Concepts: Asymptote · Rectangular hyperbola
Functions: Hyperbola