## Parametric Equations

This page covers Parametric equations.

The equation of a circle, centred at the origin, is: x^{2} + y^{2} = a^{2}, 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:

x^{2} + y^{2} = a^{2}cos^{2}q + a^{2}sin^{2}q = a^{2} .

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.

**Example**

Find the Cartesian equation given by the parametric equations:

x = at^{2} (3)

y = 2at (4)

From (4), t = y/2a

Substituting this into (3):

x = a[y/2a]^{2}

= y^{2}/4a

Hence y^{2} = 4ax