FP2 June 2005 Q8
8. The curve \(C\) which passes through \(O\) has polar equation \[r = 4a(1 + \cos\theta), \qquad -\pi \lt \theta \leqslant \pi.\] The line \(l\) has polar equation \[r = 3a\sec\theta, \qquad -\frac{\pi}{2} \lt \theta \lt \frac{\pi}{2}.\] The line \(l\) cuts \(C\) at the points \(P\) and \(Q\), as shown in the diagram.
(a) Prove that \(PQ = 6\sqrt{3}a\). (6)
The region \(R\), shown shaded in the diagram, is bounded by \(l\) and \(C\).
(b) Use calculus to find the exact area of \(R\). (7)
| Scheme | Marks |
|---|---|
| \(4a(1 + \cos\theta) = \dfrac{3a}{\cos\theta}\) or \(r = 4a\left(1 + \dfrac{3a}{r}\right)\) | M1 |
| \(4\cos^2\theta + 4\cos\theta - 3 = 0\) or \(r^2 - 4ar - 12a^2 = 0\) | A1 |
| \((2\cos\theta - 1)(2\cos\theta + 3) = 0\) or \((r - 6a)(r + 2a) = 0\) | M1 |
| \(\cos\theta = \dfrac{1}{2},\ \left(\theta = \dfrac{\pi}{3}\right)\) or \(r = 6a\) Note \(ON = 3a\) | A1 |
| \(PQ = 2 \times ON\tan\dfrac{\pi}{3} = 6\sqrt{3}a\ (*)\) | cso M1 A1 |
| or \(PQ = 2 \times \sqrt{[(6a)^2 - (3a)^2]} = 2\sqrt{(27a^2)} = 6\sqrt{3}a\ (*)\) or any complete equivalent | cso |
| (6) |
Notes
(corrected from the printed mark scheme: the first method is printed as \(PQ = 2 \times ON\tan\dfrac{\pi}{6}\); with \(\theta = \dfrac{\pi}{3}\) at \(P\) it is \(2 \times ON\tan\dfrac{\pi}{3} = 2 \times 3a \times \sqrt{3}\))
| Scheme | Marks |
|---|---|
| \(2 \times \dfrac{1}{2}\displaystyle\int_0^{\pi/3} r^2\,\mathrm{d}\theta = \ldots\int_{\ldots}^{\ldots} 16a^2(1 + \cos\theta)^2\,\mathrm{d}\theta\) \(\int r^2\,\mathrm{d}\theta\) | M1 |
| \(= \ldots\displaystyle\int_{\ldots}^{\ldots}\left(1 + 2\cos\theta + \frac{1}{2} + \frac{1}{2}\cos 2\theta\right)\mathrm{d}\theta\) \(\cos^2\theta \to \cos 2\theta\) | M1 |
| \(= \ldots\left[\dfrac{3}{2}\theta + 2\sin\theta + \dfrac{1}{4}\sin 2\theta\right]\) | A1 |
| \(= 16a^2\left[\dfrac{\pi}{2} + \sqrt{3} + \dfrac{\sqrt{3}}{8}\right]\ \ (= 2a^2[4\pi + 9\sqrt{3}] \approx 56.3a^2)\) use of their \(\dfrac{\pi}{3}\) for M1 | M1 A1 |
| Area of \(\Delta POQ = \dfrac{1}{2}6\sqrt{3}\,a \times 3a\) or \(9a^2\sqrt{3}\) | B1 |
| \(R = a^2(8\pi + 9\sqrt{3})\) | cao A1 |
| (7) | |
| (13 marks) |