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 obtained from an intersecting plane that meets tangent double cones sharing the same apex. Each cone has an axis, and the plane section is parallel to the plane formed by the axes.

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

${\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=-1}$ (which can also be written as ${\displaystyle {\frac {y^{2}}{b^{2}}}-{\frac {x^{2}}{a^{2}}}=1}$), 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.

Similarly, for a non-zero constant c, the coordinate axes form the asymptotes of the conjugate pair ${\displaystyle xy=c^{2}}$ and ${\displaystyle xy=-c^{2}}$.