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
The curve $C$ has parametric equations

$x = \dfrac{2}{t+1}$ , $\quad y = \dfrac{3t}{t+2}$ , $\quad t > 0$ .
(a) Show that a Cartesian equation of $C$ is $y = \dfrac{6 - 3x}{x + 2}$ . (3 marks)
(b) Find $\dfrac{dy}{dx}$ in terms of $t$ , and hence find the gradient of $C$ at the point where $t = 1$ . (3 marks)
(c) Find the range of values of $x$ for points on $C$ . (2 marks)
Level 4 · 8 marksTry it
2
A curve $C$ has parametric equations

$x = t^2, \qquad y = 2t, \qquad 0 \leqslant t \leqslant 3$
(a) Show that the Cartesian equation of $C$ is $y^2 = 4x$ . (2 marks)
(b) Find the area of the region bounded by $C$ , the $x$ -axis and the line $x = 9$ . (4 marks)
(c) Find an equation of the normal to $C$ at the point where $t = 2$ . (4 marks)
Level 4 · 10 marksTry it
3
The curve $C$ has parametric equations

$x = 4\sin t, \qquad y = 3\cos 2t - 1, \qquad -\dfrac{\pi}{2} \leqslant t \leqslant \dfrac{\pi}{2}$

Figure 2 shows a sketch of $C$ . The region $R$ , shown shaded, is bounded by $C$ and the $x$ -axis.Figure 2
(a) Show that the Cartesian equation of $C$ can be written as $y = a + bx^2$ , where $a$ and $b$ are rational constants to be found, and state the domain of $x$ . (4 marks)
(b) Find the exact area of $R$ . (5 marks)
Level 4 · 9 marksTry it
4
A curve has parametric equations $x = t - 1,\ y = t^2 - 2t$ . Show that the Cartesian equation of the curve is $y = x^2 - 1$ .
Level 4 · 2 marksTry it
5
A curve has parametric equations $x = t^2,\ y = t^3$ . Find a Cartesian equation of the curve in the form $y^2 = f(x)$ , and state the restriction on $x$ .
Level 4 · 3 marksTry it

6 more questions on converting to cartesian form

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