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
Hence y2 = 4ax