Converting Between Parametric and Cartesian Forms

Being able to convert between parametric equations and Cartesian equations is a key skill in the Edexcel 9MA0 specification. The conversion reveals the shape of the curve and allows you to apply techniques from other parts of the syllabus.

---

Parametric to Cartesian: Eliminating the Parameter

The general strategy is to express the parameter t (or θ) in terms of x from one equation, then substitute into the other to eliminate the parameter.

Method 1: Direct Substitution

Example: Convert x = 2t + 1, y = 3t − 2 to Cartesian form.

From the first equation: t = (x − 1)/2.

Substitute into the second: y = 3((x − 1)/2) − 2 = (3x − 3)/2 − 2 = (3x − 7)/2.

So: 2y = 3x − 7, or 3x − 2y − 7 = 0. This is a straight line.

Method 2: Using Trigonometric Identities

When the parametric equations involve sin and cos, use the identity sin²θ + cos²θ = 1.

Example: Convert x = 3cos(θ), y = 3sin(θ) to Cartesian form.

cos(θ) = x/3, sin(θ) = y/3.