A third variable, such as t, that determines both x and y.

How do you convert x=at+b, y=g(t) to Cartesian form?

Make t the subject of the x equation and substitute into y.

Cartesian form of x=a\cos\theta, y=b\sin\theta?

\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1.

How do you find where a parametric curve meets the y-axis?

Set x=0, solve for the parameter, then find y.

Is the parametric form of a curve unique?

No; different parameterisations can give the same curve.

Parametric and Cartesian formsAQA A-Level Maths: Flashcards

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What is a parameter?

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What is a parameter?: A third variable, such as $t$ , that determines both $x$ and $y$ .
How do you convert $x=at+b$ , $y=g(t)$ to Cartesian form?: Make $t$ the subject of the $x$ equation and substitute into $y$ .
Cartesian form of $x=2t+1$ , $y=t^2-3$ ?: $y=\dfrac{(x-1)^2}{4}-3$ .
Which identity converts $\cos$ and $\sin$ equations?: $\cos^2\theta+\sin^2\theta=1$ .
Cartesian form of $x=a\cos\theta$ , $y=b\sin\theta$ ?: $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ .
Cartesian form of $x=r\cos\theta$ , $y=r\sin\theta$ ?: $x^2+y^2=r^2$ , a circle.
Centre of $x=1+2\cos t$ , $y=3\sin t-2$ ?: $(1,-2)$ .
Cartesian form of $x=\frac2t$ , $y=t+1$ , with restriction?: $y=\frac2x+1$ , $x\neq0$ .
Parametric form of $y=x^2+1$ ?: $x=t$ , $y=t^2+1$ .
Parametric form of $x^2+y^2=25$ ?: $x=5\cos\theta$ , $y=5\sin\theta$ .
How do you find where a parametric curve meets the $y$ -axis?: Set $x=0$ , solve for the parameter, then find $y$ .
Greatest value of $x=1+2\cos t$ ?: $3$ , when $\cos t=1$ .
Is the parametric form of a curve unique?: No; different parameterisations can give the same curve.

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).