Conjugate hyperbola

In geometry, a conjugate hyperbola to a given hyperbola shares the same asymptotes but lies in the opposite two sectors of the plane compared to the original hyperbola.

A hyperbola and its conjugate may be constructed as conic sections derived from parallel intersecting planes and cutting tangent double cones sharing the same apex.

Using analytic geometry, the hyperbolas satisfy the symmetric equations

${\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}$ with vertices (a,0) and (–a,0), and

${\displaystyle {\frac {y^{2}}{b^{2}}}-{\frac {x^{2}}{a^{2}}}=1}$ with vertices (0,b) and (0,–b).

In case a = b they are rectangular hyperbolas, and a reflection of the plane in an asymptote exchanges the conjugates.