Hyperbola
Conjugate hyperbola

Exchange x 2 a 2 {\displaystyle {\frac {x^{2}}{a^{2}}}} and y 2 b 2 {\displaystyle {\frac {y^{2}}{b^{2}}}} to obtain the equation of the conjugate hyperbola (see diagram): y 2 b 2 − x 2 a 2 = 1 , {\displaystyle {\frac {y^{2}}{b^{2}}}-{\frac {x^{2}}{a^{2}}}=1\,} also written as x 2 a 2 − y 2 b 2 = − 1. {\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=-1\.}

A hyperbola and its conjugate may have diameters which are conjugate. In the theory of special relativity, such diameters may represent axes of time and space, where one hyperbola represents events at a given spatial distance from the center, and the other represents events at a corresponding temporal distance from the center.
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