Parametric equations
Posted in Maths, Core 4A parametric equation of a curve is one which does not give the relationship between x and y directly but rather uses a third variable, typically t, to do so. The third variable is known as the parameter. A simple example of a pair of parametric equations:
x = 5t + 3
y = t2 + 2t
Finding the equation
You need to be able to find the Cartesian equation of the curve from parametric equations, that is the equation that relates x and y directly. To do this you need to eliminate the parameter. The easiest way to do this is to rearrange on parametric equation to get the parameter as the subject and then substitute this into the other equation.
Example question: Find the equivalent Cartesian equation for these parametric equations: x = 4t y = t2 + 3
Worked solution: Firstly rearrange on equation to get t as the subject:
Then substitute this rearrangement into the second equation:
It's then just a case of tidying it up:
Parametric equation of a circle
Circles with centre (0, 0)
From Core 1, a circle with a centre of (0, 0) and radius r has the equation x2 + y2 = r2. The parametric equations of this equation are:
is the parameter of these equations.
These equations come from simple trigonometry using a triangle such as ABC in this diagram:
The x and y values are the lengths of the adjacent and hypotenuse lengths of the triangle.
Circles with centre (a, b)
A circle with an origin (a, b) has the parametric equations:
As before is the parameter of the equations.
You need to be able to recognise these as parametric equations of circles in the exam.