# graph of equation $\text{tmspace}+.1667emxy=$ constant

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

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

where $c$ is a non-zero real constant. Such a dependence between the real variables $x$ and $y$ is called an inverse proportionality (http://planetmath.org/Variation).

The graph of (1) may be inferred to be a hyperbola^{} (http://planetmath.org/Hyperbola2), because the curve has two asymptotes^{} (see asymptotes of graph of rational function) and because the form

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

of the equation is of second degree (http://planetmath.org/PolynomialRing) (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 (http://planetmath.org/Hyperbola2). 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.

$\{\begin{array}{cc}x={x}^{\prime}\mathrm{cos}{45}^{\circ}-{y}^{\prime}\mathrm{sin}{45}^{\circ}={\displaystyle \frac{{x}^{\prime}-{y}^{\prime}}{\sqrt{2}}}\hfill & \\ y={x}^{\prime}\mathrm{sin}{45}^{\circ}+{y}^{\prime}\mathrm{cos}{45}^{\circ}={\displaystyle \frac{{x}^{\prime}+{y}^{\prime}}{\sqrt{2}}}\hfill & \end{array}$ | (3) |

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

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

i.e.

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

This is recognised to be the equation of a rectangular hyperbola with the transversal axis and the conjugate axis (http://planetmath.org/Hyperbola2) on the coordinate axes.

Title | graph of equation $\text{tmspace}+.1667emxy=$ constant |
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Canonical name | GraphOfEquationxyConstant |

Date of creation | 2013-03-22 17:30:12 |

Last modified on | 2013-03-22 17:30:12 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Derivation |

Classification | msc 15-00 |

Classification | msc 51N20 |

Synonym | equation $xy=$ constant |

Related topic | Variation |

Related topic | RuledSurface |

Related topic | ExactTrigonometryTables |

Related topic | Hyperbola2 |

Related topic | UncertaintyPrinciple |

Related topic | Polytrope |