4.09b Sketch polar curves: r = f(theta)

Fig. 5 shows the curve with polar equation \(r = a(3 + 2\cos\theta)\) for \(-\pi \leqslant \theta \leqslant \pi\), where \(a\) is a constant. \includegraphics{figure_2}

1. Write down the polar coordinates of the points A and B. [2]
2. Explain why the curve is symmetrical about the initial line. [2]
3. In this question you must show detailed reasoning. Find in terms of \(a\) the exact area of the region enclosed by the curve. [4]

The diagram shows the polar curve \(C_1\) with equation \(r = 2\sin\theta\) The diagram also shows part of the polar curve \(C_2\) with equation \(r = 1 + \cos 2\theta\) \includegraphics{figure_10}

1. On the diagram above, complete the sketch of \(C_2\) [2 marks]
2. Show that the area of the region shaded in the diagram is equal to $$k\pi + m\alpha - \sin 2\alpha + q\sin 4\alpha$$ where \(\alpha = \sin^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\), and \(k\), \(m\) and \(q\) are rational numbers. [9 marks]

The equation of a curve, in polar coordinates, is $$r = \sec \theta + \tan \theta, \quad \text{for } 0 \leq \theta \leq \frac{1}{4}\pi.$$

1. Sketch the curve. [2]
2. Find the exact area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac{1}{4}\pi\). [6]

The figure below shows the curve with cartesian equation \((x^2 + y^2)^2 = xy\). \includegraphics{figure_3}

1. Show that the polar equation of the curve is \(r^2 = a \sin b\theta\), where \(a\) and \(b\) are positive constants to be determined. [3]
2. Determine the exact maximum value of \(r\). [2]
3. Determine the area enclosed by one of the loops. [4]

The equation of the curve shown on the graph is, in polar coordinates, \(r = 3\sin 2\theta\) for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\). \includegraphics{figure_1}

1. The greatest value of \(r\) on the curve occurs at the point \(P\).
   1. Show that \(\theta = \frac{1}{4}\pi\) at the point \(P\). [2]
   2. Find the value of \(r\) at the point \(P\). [1]
   3. Mark the point \(P\) on a copy of the graph. [1]
2. In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve. [5]

The curve \(S\) has polar equation \(r = 1 + \sin \theta + \sin^2 \theta\) for \(0 \leqslant \theta < 2\pi\).

1. Determine the polar coordinates of the points on \(S\) where \(\frac{dr}{d\theta} = 0\). [5]
2. Sketch \(S\). [3]

A curve has polar equation \(r = \sin \frac{1}{4}\theta\) for \(0 \leqslant \theta < 2\pi\).

1. Sketch the curve. [3]
2. Determine the area of the region enclosed by the curve. [4]

\includegraphics{figure_6} The diagram shows a sketch of the curve \(C\) with polar equation \(r = a \cos^2 \theta\), where \(a\) is a positive constant and \(-\frac{1}{2}\pi \leqslant \theta \leqslant \frac{1}{2}\pi\).

1. Explain briefly how you can tell from this form of the equation that \(C\) is symmetrical about the line \(\theta = 0\) and that the tangent to \(C\) at the pole \(O\) is perpendicular to the line \(\theta = 0\). [2]
2. The equation of \(C\) may be expressed in the form \(r = \frac{1}{2}a(1 + \cos 2\theta)\). Using this form, show that the area of the region enclosed by \(C\) is given by $$\frac{1}{16}a^2 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (3 + 4 \cos 2\theta + \cos 4\theta) \, \mathrm{d}\theta,$$ and find this area. [6]