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# graph of equation $\,xy=$ constant

Consider the equation $xy=c$, i.e.

$\displaystyle y=\frac{c}{x},$ | (1) |

where $c$ is a non-zero real constant. Such a dependence between the real variables $x$ and $y$ is called an inverse proportionality.

The graph of (1) may be inferred to be a hyperbola, because the curve has two asymptotes (see asymptotes of graph of rational function) and because the form

$\displaystyle xy-c=0$ | (2) |

of the equation is of second degree (see conic, tangent of conic section).

One can also see the graph of the equation (2) in such a coordinate system ($x^{{\prime}},\,y^{{\prime}}$) where the equation takes a canonical form of the hyperbola. The symmetry of (2) with respect to the variables $x$ and $y$ suggests to take for the new coordinate axes the axis angle bisectors $y=\pm{x}$. Therefore one has to rotate the old coordinate axes $45^{\circ}$, i.e.

$\displaystyle\begin{cases}\displaystyle x=x^{{\prime}}\cos 45^{\circ}-y^{{% \prime}}\sin 45^{\circ}=\frac{x^{{\prime}}-y^{{\prime}}}{\sqrt{2}}\\ \displaystyle y=x^{{\prime}}\sin 45^{\circ}+y^{{\prime}}\cos 45^{\circ}=\frac{% x^{{\prime}}+y^{{\prime}}}{\sqrt{2}}\end{cases}$ | (3) |

($\sin 45^{\circ}=\cos 45^{\circ}=\frac{1}{\sqrt{2}}$). Substituting (3) into (2) yields

$\frac{x^{{\prime 2}}-y^{{\prime 2}}}{2}-c=0,$ |

i.e.

$\displaystyle\frac{x^{{\prime 2}}}{2c}-\frac{y^{{\prime 2}}}{2c}=1.$ | (4) |

This is recognised to be the equation of a rectangular hyperbola with the transversal axis and the conjugate axis on the coordinate axes.

## Mathematics Subject Classification

15-00*no label found*51N20

*no label found*

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