graph of equation xy = constant∗
pahio†
2013-03-21 23:43:56
Consider the equation xy = c, i.e.
y=
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
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
(x0 , y 0 ) 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 = ±x. Therefore one has to rotate
the old coordinate axes 45◦ , i.e.

x0 − y 0


x = x0 cos 45◦ − y 0 sin 45◦ = √
2
(3)
0
x
+
y0

0
◦
0
◦

√
y
=
x
sin
45
+
y
cos
45
=

2
(sin 45◦ = cos 45◦ =
√1 ).
2
Substituting (3) into (2) yields
x02 − y 02
− c = 0,
2
∗ hGraphOfEquationxyConstanti created: h2013-03-21i by: hpahioi version: h39892i
Privacy setting: h1i hDerivationi h15-00i h51N20i
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1
i.e.
x02
y 02
−
= 1.
2c
2c
(4)
This is recognised to be the equation of a rectangular hyperbola with the
transversal axis and the conjugate axis on the coordinate axes.
2