Parametric and Cartesian formsAQA A-Level Maths: Revision notes

Section 1

Parametric equations

In parametric equations both $x$ and $y$ are given in terms of a third variable, the parameter (often $t$ or $\theta$ ). Each value of the parameter gives one point $(x,y)$ on the curve. For $x=2t+1$ , $y=t^2-3$ : $t=0$ gives $(1,-3)$ , $t=3$ gives $(7,6)$ . To find the parameter at a given point, solve one equation and then substitute into the other. To find the least or greatest value of $x$ or $y$ , work with the single equation in $t$ or $\theta$ .

Key termsparametric equationsparameter

Exam tip

Test a few values of the parameter to check the shape of the curve before you convert.

Section 2

Converting to Cartesian form by substitution

If the parameter can be isolated from one equation, make it the subject and substitute into the other. Example: $x=2t+1$ gives $t=\frac{x-1}{2}$ , so $y=\left(\frac{x-1}{2}\right)^2-3=\frac{(x-1)^2}{4}-3$ . Example: $x=\frac2t$ , $y=t+1$ gives $t=\frac2x$ and $y=\frac2x+1$ , with $x\neq0$ .

Key termsCartesian equation

Common mistake

Forgetting to square the coefficient: from $x=2t+1$ , $t^2=\frac{(x-1)^2}{4}$ , not $(x-1)^2$ .

Section 3

Converting with trigonometric identities

When the equations involve $\cos\theta$ and $\sin\theta$ , isolate each and use $\cos^2\theta+\sin^2\theta=1$ . For $x=4\cos\theta$ , $y=3\sin\theta$ : $\cos\theta=\frac x4$ , $\sin\theta=\frac y3$ , so $\frac{x^2}{16}+\frac{y^2}{9}=1$ , an ellipse. For $x=1+2\cos t$ , $y=3\sin t-2$ : $\frac{(x-1)^2}{4}+\frac{(y+2)^2}{9}=1$ , an ellipse with centre $(1,-2)$ .

Key termsellipse

Common mistake

Writing $x^2+y^2=25$ for $x=4\cos\theta$ , $y=3\sin\theta$ . The coefficients differ, so the curve is not a circle.

Section 4

Restrictions on the Cartesian form

The Cartesian equation can include points the parametric form does not, or the parametric form may leave out values. Always note the restriction. For $x=\frac2t$ , the point where $x=0$ is never reached, so the curve has $x\neq0$ . For a trigonometric pair, the range of $\cos$ and $\sin$ gives the greatest and least values: $x=1+2\cos t$ lies between $-1$ and $3$ .

Key termsrestriction

Exam tip

State the restriction on $x$ (or $y$ ) whenever the parameter cannot take all real values.

Section 5

From Cartesian to parametric form

Choose a simple parametric form and check it satisfies the equation.

$y=f(x)$ : let $x=t$ , so $y=f(t)$ . For $y=x^2+1$ , $x=t$ , $y=t^2+1$ .
Circle $x^2+y^2=25$ : $x=5\cos\theta$ , $y=5\sin\theta$ .
Ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ : $x=4\cos\theta$ , $y=3\sin\theta$ . Parametric forms are not unique; any correct pair is accepted.

Key termsparametric form

Exam tip

Check by substituting your parametric equations back into the Cartesian equation.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Parametric and Cartesian forms

A curve is given by the parametric equations $x=2t+1$ , $y=t^2-3$ , where $t$ is a real number.
Find the coordinates of the point on the curve where $y$ is least.2 marks
A curve $C$ is given by the parametric equations $x=4\cos\theta$ , $y=3\sin\theta$ for $0\le\theta<2\pi$ .
Find the coordinates of the points where $C$ meets the $y$ -axis.2 marks
A curve is given by the parametric equations $x=\dfrac{2}{t}$ , $y=t+1$ , where $t\neq0$ .
Find the Cartesian equation of the curve, and state any restriction on $x$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).