PlanetMath: graph of equation $\,xy =$ constant

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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 $$\frac{x'^2-y'^2}{2}-c = 0,$$ 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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Other names:

equation

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
There is 1 reference to this entry.

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 2037 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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