CAIE A-Level Further Mathematics 1.5.1 Polar Coordinate Conversion

Practise replacing x, y and r consistently to convert whole curve equations between Cartesian and polar form, while respecting any stated coordinate interval.

Exam points

replace x and y by their polar expressions and use the radial identity to simplify the equation
when eliminating the polar variables, rearrange to a Cartesian equation with no residual fractions

[Maximum number: 4]

Show that the curve with Cartesian equation

\left(x^{2}+y^{2}\right)^{\frac{5}{2}}=4 x y\left(x^{2}-y^{2}\right)

has polar equation $r=\sin 4 \theta$ .

The curve C has polar equation $r=\sin 4 \theta$ , for $0 \leqslant \theta \leqslant \frac{1}{4} \pi$ .

$r^{5}=4 x y\left(x^{2}-y^{2}\right)$

B1

Uses $r^{2}=x^{2}+y^{2}$ .

$r^{5}=4 r^{4} \cos \theta \sin \theta\left(\cos ^{2} \theta-\sin ^{2} \theta\right)$

B1

Uses $x=r \cos \theta$ and $y=r \sin \theta$ .

$r=2 \sin 2 \theta \cos 2 \theta$

M1

Applies at least one double angle formula.

$r=\sin 4 \theta$

A1

Applies both double angle formulae, AG.

4

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1.5.1—Polar coordinates question 1