A-level · Coordinate geometry

Converting to Cartesian formExam-style questions

Eliminating the parameter to find the Cartesian equation of a parametric curve, by substitution or rearrangement, and carrying the domain restriction from the parameter through to the Cartesian equation.

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23 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
A curve $C$ has parametric equations

$x = 2\cos t$ , $\quad y = \cos 2t$ , $\quad 0 \le t \le \pi$ .
(a) Show that $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 2\cos t$ . (3 marks)
(b) Find the Cartesian equation of $C$ , stating the range of values of $x$ for which the curve is defined. (3 marks)
(c) Verify that differentiating your Cartesian equation gives a result consistent with part (a). (1 mark)
Level 3 · 7 marksTry it
2
A curve has parametric equations $x = t + 1,\ y = \dfrac{1}{t}$ , $t \ne 0$ . Find a Cartesian equation of the curve and state any restriction on $x$ .
Level 3 · 3 marksTry it
3
A curve has parametric equations $x = \sqrt{t},\ y = t - 4$ for $t \ge 0$ . Find the Cartesian equation of the curve and state its domain.
Level 3 · 3 marksTry it
4
A curve has parametric equations $x = \dfrac{2}{t},\ y = t + 1$ , $t \ne 0$ . Find a Cartesian equation of the curve and state any restriction on $x$ .
Level 3 · 3 marksTry it
5
A curve has parametric equations $x = \sqrt{t - 1}, \; y = 3t$ for $t \geqslant 1$ .

Find a Cartesian equation of the curve, stating the range of valid $x$ -values. (4 marks)
Level 3 · 4 marksTry it

2 more questions on converting to cartesian form

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