graph of equation \tmspace+.1667emxy= constant

Consider the equation  xy=c,  i.e.

y=cx, (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 hyperbolaMathworldPlanetmathPlanetmath (, because the curve has two asymptotesMathworldPlanetmath (see asymptotes of graph of rational function) and because the form

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 systemMathworldPlanetmath (x,y) where the equation takes a canonical form of the hyperbola ( The symmetryMathworldPlanetmath of (2) with respect to the variables x and y suggests to take for the new coordinate axes the axis angle bisectorsMathworldPlanetmathy=±x. Therefore one has to rotate the old coordinate axes 45, i.e.

{x=xcos45-ysin45=x-y2y=xsin45+ycos45=x+y2 (3)

(sin45=cos45=12). Substituting (3) into (2) yields



x22c-y22c=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.

Title graph of equation \tmspace+.1667emxy= constant
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