conjugate hyperbola

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The simplest form of the equation presenting a hyperbola (without the mixed $xy$ -term) in a rectangular coordinate system is got when the coordinate axes coincide with the principal axes of the hyperbola, and it has the form

\displaystyle\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1.

(1)

Here, $a\,(>0)$ is the length of the transverse semiaxis and $b\,(>0)$ the length of the conjugate semiaxis of the hyperbola.

The equation

\displaystyle\frac{y^{2}}{b^{2}}-\frac{x^{2}}{a^{2}}=1

(2)

\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1.

presents the conjugate hyperbola of (1). Its transverse axis is the conjugate axis of (1) and its conjugate axis the transverse axis of (1). Both hyperbolas are conjugate hyperbolas of each other. They have the common asymptotes

\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0

and their foci are on the circle $x^{2}\!+\!y^{2}=a^{2}\!+\!b^{2}$ .

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