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[parent] graph of equation $\,xy =$ constant (Derivation)

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',\,y'$) 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.

\begin{align*}\begin{cases}\displaystyle x = x'\cos45^\circ-y'\sin45^\circ = \fr... ...= x'\sin45^\circ+y'\cos45^\circ = \frac{x'+y'}{\sqrt{2}} \end{cases}\end{align*} (3)

( $ \sin45^\circ = \cos45^\circ = \frac{1}{\sqrt{2}}$). Substituting (3) into (2) yields
$\displaystyle \frac{x'^2-y'^2}{2}-c = 0,$
i.e.
$\displaystyle \frac{x'^2}{2c}-\frac{y'^2}{2c} = 1.$ (4)

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



"graph of equation $\,xy =$ constant" is owned by pahio.
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See Also: variation, ruled surface, exact trigonometry tables, hyperbola

Other names:  equation $xy =$ constant
Keywords:  rectangular hyperbola

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Cross-references: transversal axis, rectangular hyperbola, rotate, angle bisectors, axis, coordinate, symmetry, canonical, coordinate system, tangent of conic section, conic, asymptotes of graph of rational function, asymptotes, curve, graph, variables, real, equation
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This is version 6 of graph of equation $\,xy =$ constant, born on 2007-08-25, modified 2008-01-06.
Object id is 9892, canonical name is GraphOfEquationXyConstant.
Accessed 1108 times total.

Classification:
AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)
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