A-Level — mathsAdvanced2 min read

Parametric Equations

Work with parametric curves at A-Level. Convert between parametric and Cartesian forms and find intersections.

Tutor AI Team2026-02-25T01:55:14.769Z

Parametric equations express $x$ and $y$ separately in terms of a parameter $t$ (or $\theta$ ). This allows representation of curves that cannot be expressed as $y = f(x)$ .

Core Concepts

Converting to Cartesian

Eliminate the parameter.

$x = 2t$ , $y = t^2$ → $t = \frac{x}{2}$ → $y = \frac{x^2}{4}$ .

Trig Parametric Equations

$x = \cos\theta$ , $y = \sin\theta$ → $x^2 + y^2 = 1$ (circle).

$x = 2\cos\theta$ , $y = 3\sin\theta$ → $\frac{x^2}{4} + \frac{y^2}{9} = 1$ (ellipse).

Finding Points

Substitute specific $t$ values to find coordinates.

Worked Example: Example 1

Problem

$x = t + \frac{1}{t}$ , $y = t - \frac{1}{t}$ .

$x + y = 2t$ , $x - y = \frac{2}{t}$ . $(x+y)(x-y) = 4$ → $x^2 - y^2 = 4$ .

Solution

Worked Example: Example 2

Problem

Where does $x = 3t$ , $y = \frac{6}{t}$ meet $y = x - 4$ ?

$\frac{6}{t} = 3t - 4$ → $6 = 3t^2 - 4t$ → $3t^2 - 4t - 6 = 0$ . Solve for $t$ , then find $(x,y)$ .

Solution

Practice Problems

1. Convert $x = 5\cos\theta$ , $y = 5\sin\theta$ to Cartesian form.
1. Find where $x = t^2$ , $y = 2t$ meets $y = x - 3$ .

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Key Takeaways

✓
Eliminate the parameter to get Cartesian form.
✓
Trig: use $\sin^2\theta + \cos^2\theta = 1$ .
✓
Substitute $t$ values to find specific points.

Tags:

#a-level#maths#pure#parametric#curves

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Core Concepts

Converting to Cartesian

Trig Parametric Equations

Finding Points

Worked Example: Example 1

Worked Example: Example 2

Practice Problems

Key Takeaways

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