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.

Example

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