CAIE A-Level Further Math ASematics 1.5.1 Polar Coordinates

[Maximum number: 6]

(a)

Show that the curve with Cartesian equation

\left(x-\frac{1}{2}\right)^{2}+y^{2}=\frac{1}{4}

has polar equation $r=\cos \theta$ .

The curves $C_{1}$ and $C_{2}$ have polar equations

r=\cos \theta \quad \text { and } \quad r=\sin 2 \theta

respectively, where $0 \leqslant \theta \leqslant \frac{1}{2} \pi$ . The curves $C_{1}$ and $C_{2}$ intersect at the pole and at another point P.

[ 3 ]

$x^{2}-x+\frac{1}{4}+y^{2}=\frac{1}{4} \Rightarrow r^{2}-r \cos \theta+\frac{1}{4}=\frac{1}{4}$
Uses $x^{2}+y^{2}=r^{2}$ and $x=r \cos \theta$ .

$r(r-\cos \theta)=0$
Factorises.

$[r \neq 0 \Rightarrow] r=\cos \theta$
AG.

3

(b)

Find the polar coordinates of P.

[ 3 ]

$\sin 2 \theta=\cos \theta \Rightarrow 2 \sin \theta \cos \theta=\cos \theta$
Sets r values equal and uses $\sin 2 \theta=2 \sin \theta \cos \theta$ .

$\cos \theta \neq 0 \Rightarrow \sin \theta=\frac{1}{2}$ $\cos \theta \neq 0$ must be recognised.

$\left(\frac{1}{2} \sqrt{3}, \frac{1}{6} \pi\right)$
3

CAIE A-Level Further Math AS 1.5.1 Polar Coordinates Questions

Exam points

CAIE A-Level Further Math AS 1.5.1 Polar Coordinates Questions question 1

Question (a)

Question (b)