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

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 ${\displaystyle xy=-c^{2}}$.

Apollonius of Perga introduced the conjugate hyperbola through a geometric construction: "Given two straight lines bisecting one another at any angle, to describe two hyperbolas each with two branches such that the straight lines are conjugate diameters of both hyperbolas." "The two hyperbolas so constructed are called conjugate hyperbolas, and [the] last drawn is the hyperbola conjugate to the first."

The following property was described by Apollonius: let PP', DD' be conjugate diameters of two conjugate hyperbolas, Draw the tangents at P, P', D, D'. Then ... the tangents form a parallelogram, and the diagonals of it, LM, L'M', pass through the center [C]. Also PL = PL' = P'M = P'M' = CD. It is noted that the diagonals of the parallelogram are the asymptotes common to both hyperbolas. Either PP' or DD' is a transverse diameter, with the opposite one being the conjugate diameter.

Elements of Dynamic (1878) by W. K. Clifford identifies the conjugate hyperbola.