Parametric Equations - Eliminating the Parameter (AQA A-Level Mathematics): Revision Notes

1. Identify the parametric equations: These are usually given in the form:

$x = f(t)$

$y = g(t)$

2. Solve one equation for the parameter $t$ : Choose the simpler of the two equations to solve for $t$ . Suppose we have $x = f(t)$, then solve for $t$ in terms of $x$:

$t = f^{-1}(x)$

(Note: $f^{-1}$ denotes the inverse function of $f$, if it exists.) 3. Substitute into the other equation: Replace $t$ in the equation $y = g(t)$ with the expression you found for $t$ in terms of $x$ :

$y = g(f^{-1}(x))$

This gives the Cartesian equation relating $x$ and $y$.

📑Example:

Consider the parametric equations:

$x = 2t + 3$

$y = t^2 - 1$

Step 1: Solve $x = 2t + 3$ for $t$:

$x - 3 = 2t$

$t = \frac{x - 3}{2}$

Step 2: Substitute $t = \frac{x - 3}{2}$ into the second equation $y = t^2 - 1$ :

$y = \left(\frac{x - 3}{2}\right)^2 - 1$

Step 3: Simplify the expression:

$y = \frac{(x - 3)^2}{4} - 1$

$y = \frac{(x - 3)^2 - 4}{4}$

$y = \frac{(x - 3)^2}{4} - 1$

So, the Cartesian equation is:

$y = \frac{(x - 3)^2}{4} - 1$

This equation represents the relationship between $x$ and $y$ without involving the parameter $t$ .

🤔Tips:

• If the parametric equations involve trigonometric functions like $\sin t$ and $\cos t$ , use identities such as $\sin^2 t + \cos^2 t = 1$ to eliminate $t$ .
• Some parametric forms may be easier to eliminate than others. If the algebra becomes too complicated, double-check the work or try a different approach. Understanding how to eliminate the parameter will help you move between parametric and Cartesian forms, which is a valuable skill for solving various problems in A Level Maths.

📑Example: Write $x = \cos \theta$ and $y = \sin \theta$ with $0 \leq \theta \leq 2\pi$ in Cartesian form.

📑Example: Write $x = 5 \cos t + 1$ and $y = 6 \sin t$ with $0 \leq t \leq \pi$ in Cartesian form.

Square both:

Add:

This gives: