Rectangular hyperbola referred to its asymptotes

If we choose the asymptotes themselves of the rectangular hyperbola as the Cartesian axes, the equation simplifies further.

Property — Rectangular hyperbola referred to its asymptotes

If we rotate the reference frame by $45°$ so that the asymptotes coincide with the Cartesian axes, the rectangular hyperbola $x^2 - y^2 = a^2$ takes the form $xy = k, \qquad k = \tfrac{a^2}{2}.$

In this frame, the hyperbola is the graph of the function $y = \tfrac{k}{x}$: what in Year One we called “inverse proportionality”. The same curve, therefore, appears as $x^2-y^2=a^2$ in the frame centred on the focal axis and as $xy=k$ in the frame aligned with the asymptotes.

Links

Topics: Homographic hyperbola
Concepts: Asymptote · Rectangular hyperbola · Inverse proportionality
Functions: Hyperbola
Skills: Analytic geometry