Polar coordinates

WFM02Topic 2.73 questions

Polar curves, areas enclosed by polar curves, and tangents parallel or perpendicular to the initial line.

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Original practice questions written in the style of the exam.

Question 15 marks
Find the exact area enclosed by the curve with polar equation $r = 2 + \cos\theta$ , $0 \leq \theta < 2\pi$ .
Mark scheme
- M1 Area $= \dfrac{1}{2}\displaystyle\int_0^{2\pi} (2 + \cos\theta)^2\,d\theta$ (1)
Question 25 marks
The curve $C$ has polar equation $r = 1 + \cos\theta$ , $-\pi < \theta \leq \pi$ .

2π

(

cosθ)2dθ

A1 Integrand

4 + 4\cos\theta + \cos^2\theta

(1)

M1 Uses

\cos^2\theta = \tfrac{1}{2}(1 + \cos 2\theta)

(1)

\dfrac{1}{2}\left[\dfrac{9}{2}\theta + 4\sin\theta + \dfrac{1}{4}\sin 2\theta\right]_0^{2\pi}

(1)

\dfrac{9\pi}{2}

(1)

≤

Find the polar coordinates of the points on $C$ , other than the pole, where the tangent is parallel to the initial line.

Mark scheme

M1 $y = r\sin\theta = \sin\theta + \sin\theta\cos\theta$ (1)
M1 $\dfrac{dy}{d\theta} = \cos\theta + \cos^2\theta - \sin^2\theta$ (1)
A1 Sets to zero: $2\cos^2\theta + \cos\theta - 1 = 0$ (1)
A1 $(2\cos\theta - 1)(\cos\theta + 1) = 0$ ; $\cos\theta = -1$ gives the pole, so (1)
A1 $\left(\tfrac{3}{2}, \tfrac{\pi}{3}\right)$ and (1)

−

\cos\theta = \tfrac{1}{2}

)

\left(\tfrac{3}{2}, -\tfrac{\pi}{3}\right)