graph of equation $+.1667emxy=$ 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.

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

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