Parametric Equations

This page covers Parametric equations.

The equation of a circle, centred at the origin, is: x2 + y2 = a2, where a is the radius.

Suppose we have a curve which is described by the following two equations:

x = acosq    (1)
y = asinq     (2)

We can eliminate q by squaring and adding the two equations:

x2 + y2 = a2cos2q + a2sin2q = a2 .

Hence equations (1) and (2) together also represent a circle centred at the origin with radius a and are known as the parametric equations of the circle. q is known as the parameter. As q varies between 0 and 2p, x and y vary.

It is often useful to have the parametric representation of a particular curve. The normal Cartesian representation (in terms of x's and y's) can be obtained by eliminating the parameter as above.


Find the Cartesian equation given by the parametric equations:

x = at2     (3)
y = 2at     (4)

From (4), t = y/2a

Substituting this into (3):
x = a[y/2a]2
= y2/4a
Hence y2 = 4ax