# graph of equation $\tmspace+{.1667em}xy=$ 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 (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

 $\displaystyle xy-c=0$ (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.

 $\displaystyle\begin{cases}\displaystyle x=x^{\prime}\cos 45^{\circ}-y^{\prime}% \sin 45^{\circ}=\frac{x^{\prime}-y^{\prime}}{\sqrt{2}}\\ \displaystyle y=x^{\prime}\sin 45^{\circ}+y^{\prime}\cos 45^{\circ}=\frac{x^{% \prime}+y^{\prime}}{\sqrt{2}}\end{cases}$ (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 (http://planetmath.org/Hyperbola2) on the coordinate axes.

Title graph of equation $\tmspace+{.1667em}xy=$ constant GraphOfEquationxyConstant 2013-03-22 17:30:12 2013-03-22 17:30:12 pahio (2872) pahio (2872) 9 pahio (2872) Derivation msc 15-00 msc 51N20 equation $xy=$ constant Variation RuledSurface ExactTrigonometryTables Hyperbola2 UncertaintyPrinciple Polytrope