A-level · Coordinate geometry

Parametric curves and trig identitiesExam-style questions

Converting trigonometric parametric equations to Cartesian form using identities — sin²+cos²≡1 for circles and ellipses, sec²−tan²≡1, and the double-angle formulae — and recognising the curve that results.

Start practising Read the lesson

16 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 has parametric equations $x = \cos t,\ y = \sin 2t$ . Show that a Cartesian equation of the curve is $y^2 = 4x^2(1 - x^2)$ .
Level 5 · 3 marksTry it
2
A curve has parametric equations $x = \sin t, \; y = \sin\left(t + \dfrac{\pi}{3}\right)$ for $0 \leqslant t \leqslant \dfrac{\pi}{2}$ .

Show that a Cartesian equation of the curve is

$y = \dfrac{x}{2} + \dfrac{\sqrt{3}}{2}\sqrt{1 - x^{2}}$ (5 marks)
Level 5 · 5 marksTry it
3
The curve $C$ has parametric equations

$x = 2 + 2\cos 2t \qquad\qquad y = 2 + 2\sin t \qquad\qquad -\frac{\pi}{2} \leqslant t \leqslant \frac{\pi}{6}$

At the point $P$ on $C$ , $t = -\dfrac{\pi}{6}$ . The line $l$ is the normal to $C$ at $P$ .
(a) Use parametric differentiation to show that $l$ has equation

$y = 7 - 2x$

(5 marks)
(b) Show that every point of $C$ lies on the curve with equation

$x = 4 - (y - 2)^2$

(2 marks)

The line $l_1$ has equation $y = k - 2x$ , where $k$ is a constant.
(c) Find the set of values of $k$ for which $l_1$ meets $C$ at two distinct points.

(5 marks)
Level 5 · 12 marksTry it