9. Parametric Equations (Edexcel A Level Maths: Pure)
Revision Note
Parametric Equations
Understanding Parametric Equations
Parametric equations are a way to represent the coordinates of points on a curve using a set of equations. These equations express the coordinates in terms of one or more parameters, typically denoted by 't'. By varying the parameter 't', we can trace out the curve.
A set of parametric equations can be written as:
where f(t) and g(t) are functions of the parameter 't'.
Example 1:
A classic example of a parametric equation is the representation of a circle:
Here, 'a' is the radius of the circle, and 't' varies from 0 to 2Ï€. As 't' changes, the values of x and y trace out a circle with radius 'a'.
Eliminating the Parameter
To find the Cartesian equation of a curve from its parametric equations, you need to eliminate the parameter 't'. This can be done by expressing 't' in terms of 'x' or 'y' and substituting it back into one of the parametric equations.
Example 2:
Consider the following parametric equations:
To eliminate 't', solve the first equation for 't':
Substitute this expression for 't' into the second equation:
Simplify the equation to obtain the Cartesian equation:
More information: Eliminating the parameter of parametric equations
Differentiation and Integration of Parametric Equations
To differentiate or integrate a parametric equation, you need to use the chain rule. This involves differentiating both the 'x' and 'y' equations with respect to 't' and then using these derivatives to find or the integral of y with respect to x.
Example 3:
Differentiate the parametric equations from Example 2:
To find , divide by :
Example 4:
Integrating the parametric equations from Example 2:
Parametric Equation Exam Tips
- Be familiar with common parametric representations (for example, a circle mentioned above)
- Trigonometric identities can often help to eliminate parameters
- Pay attention to any specific domain of 't' which determines the range of values it can take
Parametric Equations Practice Questions
To help you master parametric equations, try these parametric equation topic questions.