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:

table row x equals cell straight f open parentheses t close parentheses end cell row y equals cell straight g open parentheses t close parentheses end cell end table

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:

table row x equals cell a space cos space t end cell row y equals cell a space sin space t end cell end table

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:

table row x equals cell 2 t plus 1 end cell row y equals cell t squared minus 1 end cell end table

To eliminate 't', solve the first equation for 't':

t equals fraction numerator x minus 1 over denominator 2 end fraction

Substitute this expression for 't' into the second equation:

y equals open parentheses fraction numerator x minus 1 over denominator 2 end fraction close parentheses squared minus 1

Simplify the equation to obtain the Cartesian equation:

table row y equals cell fraction numerator x squared minus 2 x plus 1 over denominator 4 end fraction minus 1 end cell row blank equals cell 1 fourth x to the power of 2 space end exponent minus 1 half x minus 3 over 4 end cell end table

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 fraction numerator straight d y over denominator straight d x end fraction or the integral of y with respect to x.

Example 3:

Differentiate the parametric equations from Example 2:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator straight d x over denominator dt end fraction end cell equals 2 row cell dy over dt end cell equals cell 2 t end cell end table

To find fraction numerator straight d y over denominator straight d x end fraction, divide fraction numerator straight d y over denominator straight d t end fraction by fraction numerator straight d x over denominator straight d t end fraction:

fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 2 t over denominator 2 end fraction equals t

Example 4:
Integrating the parametric equations from Example 2:

table row cell integral y space straight d x end cell equals cell integral y fraction numerator straight d x over denominator straight d t end fraction straight d t end cell row blank equals cell integral open parentheses t squared minus 1 close parentheses cross times 2 space straight d t end cell row blank equals cell 2 over 3 t to the power of 3 space end exponent minus 2 t plus c end cell end table

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.