4.09c Area enclosed: by polar curve

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with polar equation $$r = a + 3\cos \theta, \quad a > 0, \quad 0 \leq \theta < 2\pi.$$ The area enclosed by the curve is \(\frac{10\pi}{2}\). Find the value of \(a\). [8]

\includegraphics{figure_1} Figure 1 Figure 1 shows the curves given by the polar equations $$r = 2, \quad 0 \leq \theta \leq \frac{\pi}{2},$$ and $$r = 1.5 + \sin 3\theta, \quad 0 \leq \theta \leq \frac{\pi}{2}.$$

1. Find the coordinates of the points where the curves intersect. [3]

The region \(S\), between the curves, for which \(r > 2\) and for which \(r < (1.5 + \sin 3\theta)\), is shown shaded in Figure 1.

1. Find, by integration, the area of the shaded region \(S\), giving your answer in the form \(a\pi + b\sqrt{3}\), where \(a\) and \(b\) are simplified fractions. [7]

\includegraphics{figure_1} Figure 1 shows a closed curve \(C\) with equation $$r = 3(\cos 2\theta)^{\frac{1}{2}}, \quad \text{where } -\frac{\pi}{4} \leq \theta \leq \frac{\pi}{4}, \frac{3\pi}{4} \leq \theta \leq \frac{5\pi}{4}.$$ The lines \(PQ\), \(SR\), \(PS\) and \(QR\) are tangents to \(C\), where \(PQ\) and \(SR\) are parallel to the initial line and \(PS\) and \(QR\) are perpendicular to the initial line. The point \(O\) is the pole.

1. Find the total area enclosed by the curve \(C\), shown unshaded inside the rectangle in Figure 1. [4]
2. Find the total area of the region bounded by the curve \(C\) and the four tangents, shown shaded in Figure 1. [9]

The diagram above shows the curve \(C_1\) which has polar equation \(r = a(3 + 2\cos\theta)\), \(0 \leq \theta < 2\pi\) and the circle \(C_2\) with equation \(r = 4a\), \(0 \leq \theta < 2\pi\), where \(a\) is a positive constant.

1. Find, in terms of \(a\), the polar coordinates of the points where the curve \(C_1\) meets the circle \(C_2\).(4)

The regions enclosed by the curves \(C_1\) and \(C_2\) overlap and this common region \(R\) is shaded in the figure.

1. Find, in terms of \(a\), an exact expression for the area of the region \(R\).(8)
2. In a single diagram, copy the two curves in the diagram above and also sketch the curve \(C_3\) with polar equation \(r = 2a\cos\theta\), \(0 \leq \theta < 2\pi\) Show clearly the coordinates of the points of intersection of \(C_1\), \(C_2\) and \(C_3\) with the initial line, \(\theta = 0\).(3)(Total 15 marks)

\includegraphics{figure_4}

The curve \(C\) shown in the diagram above has polar equation $$r = 4(1 - \cos\theta), 0 \leq \theta \leq \frac{\pi}{2}.$$ At the point \(P\) on \(C\), the tangent to \(C\) is parallel to the line \(\theta = \frac{\pi}{2}\).

1. Show that \(P\) has polar coordinates \(\left(2, \frac{\pi}{3}\right)\).(5)

The curve \(C\) meets the line \(\theta = \frac{\pi}{2}\) at the point \(A\). The tangent to \(C\) at the initial line at the point \(N\). The finite region \(R\), shown shaded in the diagram above, is bounded by the initial line, the line \(\theta = \frac{\pi}{2}\), the arc \(AP\) of \(C\) and the line \(PN\).

1. Calculate the exact area of \(R\). (8)

\includegraphics{figure_8}

The curve \(C\) has polar equation \(r = 3a \cos \theta\), \(-\frac{\pi}{2} \leq \frac{\pi}{2}\). The curve \(D\) has polar equation \(r = a(1 + \cos \theta)\), \(-\pi \leq \theta < \pi\). Given that \(a\) is a positive constant,

1. sketch, on the same diagram, the graphs of \(C\) and \(D\), indicating where each curve cuts the initial line. [4] The graphs of \(C\) intersect at the pole \(O\) and at the points \(P\) and \(Q\).
2. Find the polar coordinates of \(P\) and \(Q\). [3]
3. Use integration to find the exact value of the area enclosed by the curve \(D\) and the lines \(\theta = 0\) and \(\theta = \frac{\pi}{3}\). [7] The region \(R\) contains all points which lie outside \(D\) and inside \(C\). Given that the value of the smaller area enclosed by the curve \(C\) and the line \(\theta = \frac{\pi}{3}\) is $$\frac{3a^2}{16}(2\pi - 3\sqrt{3}),$$
4. show that the area of \(R\) is \(\pi a^2\). [4]

\includegraphics{figure_1} The curve \(C\) shown in Fig. 1 has polar equation $$r = a(3 + \sqrt{5} \cos \theta), \quad -\pi \leq \theta < \pi$$

1. Find the polar coordinates of the points \(P\) and \(Q\) where the tangents to \(C\) are parallel to the initial line. [6] The curve \(C\) represents the perimeter of the surface of a swimming pool. The direct distance from \(P\) to \(Q\) is \(20\) m.
2. Calculate the value of \(a\). [3]
3. Find the area of the surface of the pool. [6]

\includegraphics{figure_1} Figure 1 shows a sketch of the cardioid \(C\) with equation \(r = a(1 + \cos \theta)\), \(-\pi < \theta \leq \pi\). Also shown are the tangents to \(C\) that are parallel and perpendicular to the initial line. These tangents form a rectangle \(WXYZ\).

1. Find the area of the finite region, shaded in Fig. 1, bounded by the curve \(C\). [6]
2. Find the polar coordinates of the points \(A\) and \(B\) where \(WZ\) touches the curve \(C\). [5]
3. Hence find the length of \(WX\). [2] Given that the length of \(WZ\) is \(\frac{3\sqrt{3}a}{2}\),
4. find the area of the rectangle \(WXYZ\). [1] A heart-shape is modelled by the cardioid \(C\), where \(a = 10\) cm. The heart shape is cut from the rectangular card \(WXYZ\), shown in Fig. 1.
5. Find a numerical value for the area of card wasted in making this heart shape. [2]

\includegraphics{figure_1} A logo is designed which consists of two overlapping closed curves. The polar equations of these curves are $$r = a(3 + 2\cos \theta) \quad \text{and}$$ $$r = a(5 - 2 \cos \theta), \quad 0 \leq \theta < 2\pi.$$ Figure 1 is a sketch (not to scale) of these two curves.

1. Write down the polar coordinates of the points \(A\) and \(B\) where the curves meet the initial line. [2]
2. Find the polar coordinates of the points \(C\) and \(D\) where the two curves meet. [4]
3. Show that the area of the overlapping region, which is shaded in the figure, is $$\frac{a^2}{3}(49\pi - 48\sqrt{3}).$$ [8]

\includegraphics{figure_1} Figure 1 is a sketch of the two curves \(C_1\) and \(C_2\) with polar equations $$C_1 : r = 3a(1 - \cos \theta), \quad -\pi \leq \theta < \pi$$ and $$C_2 : r = a(1 + \cos \theta), \quad -\pi \leq \theta < \pi.$$ The curves meet at the pole \(O\), and at the points \(A\) and \(B\).

1. Find, in terms of \(a\), the polar coordinates of the points \(A\) and \(B\). [4]
2. Show that the length of the line \(AB\) is \(\frac{3\sqrt{3}}{2}a\). [2] The region inside \(C_2\) and outside \(C_1\) is shown shaded in Fig. 1.
3. Find, in terms of \(a\), the area of this region. [7] A badge is designed which has the shape of the shaded region. Given that the length of the line \(AB\) is \(4.5\) cm,
4. calculate the area of this badge, giving your answer to three significant figures. [3]

\includegraphics{figure_1} The curve \(C\) which passes through \(O\) has polar equation $$r = 4a(1 + \cos \theta), \quad -\pi < \theta \leq \pi$$ The line \(l\) has polar equation $$r = 3a \sec \theta, \quad -\frac{\pi}{2} < \theta < \frac{\pi}{2}.$$ The line \(l\) cuts \(C\) at the points \(P\) and \(Q\), as shown in Figure 1.

1. Prove that \(PQ = 6\sqrt{3}a\). [6] The region \(R\), shown shaded in Figure 1, is bounded by \(l\) and \(C\).
2. Use calculus to find the exact area of \(R\). [7]

\includegraphics{figure_7} The diagram shows the curve with equation, in polar coordinates, $$r = 3 + 2\cos \theta, \quad \text{for } 0 \leq \theta < 2\pi.$$ The points \(P\), \(Q\), \(R\) and \(S\) on the curve are such that the straight lines \(POR\) and \(QOS\) are perpendicular, where \(O\) is the pole. The point \(P\) has polar coordinates \((r, \alpha)\).

1. Show that \(OP + OQ + OR + OS = k\), where \(k\) is a constant to be found. [3]
2. Given that \(\alpha = \frac{1}{4}\pi\), find the exact area bounded by the curve and the lines \(OP\) and \(OQ\) (shaded in the diagram). [5]

The equation of a curve, in polar coordinates, is $$r = e^{-2\theta}, \quad \text{for } 0 \leq \theta \leq \pi.$$

1. Sketch the curve, stating the polar coordinates of the point at which \(r\) takes its greatest value. [2]
2. The pole is \(O\) and points \(P\) and \(Q\), with polar coordinates \((r_1, \theta_1)\) and \((r_2, \theta_2)\) respectively, lie on the curve. Given that \(\theta_2 > \theta_1\), show that the area of the region enclosed by the curve and the lines \(OP\) and \(OQ\) can be expressed as \(k(r_1^2 - r_2^2)\), where \(k\) is a constant to be found. [5]

\includegraphics{figure_8} The diagram shows two curves, \(C_1\) and \(C_2\), which intersect at the pole \(O\) and at the point \(P\). The polar equation of \(C_1\) is \(r = \sqrt{2}\cos\theta\) and the polar equation of \(C_2\) is \(r = \sqrt{2}\sin 2\theta\). For both curves, \(0 \leq \theta \leq \frac{1}{2}\pi\). The value of \(\theta\) at \(P\) is \(\alpha\).

1. Show that \(\tan\alpha = \frac{1}{2}\). [2]
2. Show that the area of the region common to \(C_1\) and \(C_2\), shaded in the diagram, is \(\frac{1}{4}\pi - \frac{1}{2}\alpha\). [7]

Roberto is solving this mathematics problem: Roberto's solution is as follows:

1. Sketch the curve \(C_1\) [2 marks]
2. Explain what Roberto has done wrong. [2 marks]
3. Find the area enclosed by \(C_1\) [2 marks]
4. \(P\) and \(Q\) are distinct points on \(C_1\) for which \(r\) is a maximum. \(P\) is above the initial line. Find the polar coordinates of \(P\) and \(Q\) [2 marks]
5. The matrix \(\mathbf{M} = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}\) represents the transformation T T maps \(C_1\) onto a curve \(C_2\)
   1. T maps \(P\) onto the point \(P'\) Find the polar coordinates of \(P'\) [4 marks]
   2. Find the area enclosed by \(C_2\) Fully justify your answer. [2 marks]

The curve \(C\) has polar equation \(r = 2 + \tan \theta\) The curve \(C\) meets the line \(\theta = \frac{\pi}{4}\) at the point \(A\) The point \(B\) has polar coordinates \((4, 0)\) The diagram shows part of the curve \(C\), and the points \(A\) and \(B\) \includegraphics{figure_16}

1. Show that the area of triangle \(OAB\) is \(3\sqrt{2}\) units. [2 marks]
2. Find the area of the shaded region. Give your answer in an exact form. [7 marks]

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_14}

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 diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line. \includegraphics{figure_11} The polar equation of \(C\) is \(r = 4 + 2\cos \theta\), \quad \(-\pi \leq \theta \leq \pi\)

1. Show that the area of the region bounded by the curve \(C\) is \(18\pi\) [4 marks]
2. Points \(A\) and \(B\) lie on the curve \(C\) such that \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\) and \(AOB\) is an equilateral triangle. Find the polar equation of the line segment \(AB\) [4 marks]

The diagram below shows the curve with polar equation \(r = \sin 3\theta\) for \(0 \leqslant \theta \leqslant \frac{1}{3}\pi\). \includegraphics{figure_7}

1. Find the values of \(\theta\) at the pole. [1]
2. Find the polar coordinates of the point on the curve where \(r\) takes its maximum value. [2]
3. In this question you must show detailed reasoning. Find the exact area enclosed by the curve. [4]
4. Given that \(\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta\), find a cartesian equation for the curve. [2]

In polar coordinates, the equation of a curve, \(C\), is \(r = 6\sin(2\theta)\sinh\left(\frac{1}{3}\theta\right)\) for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\). The pole of the polar coordinate system corresponds to the origin of the cartesian system and the initial line corresponds to the positive \(x\)-axis.

1. Explain how you can tell that \(C\) comprises a single loop in the first quadrant, passing through the pole. [3]

The incomplete table below shows values of \(r\) for various values of \(\theta\).

\(\theta\)	0	\(\frac{1}{12}\pi\)	\(\frac{1}{6}\pi\)	\(\frac{1}{4}\pi\)	\(\frac{1}{3}\pi\)	\(\frac{5}{12}\pi\)	\(\frac{1}{2}\pi\)
\(r\)	0	0.262			1.851

1. Use the copy of the table and the polar coordinate system diagram given in the Printed Answer Booklet to complete the table and sketch \(C\). [3]

The point on \(C\) which is furthest away from the pole is denoted by \(A\) and the value of \(\theta\) at \(A\) is denoted by \(\phi\).

1. Show that \(\phi\) satisfies the equation \(\phi = \frac{3}{4}\ln\left(\frac{6-\tan 2\phi}{6+\tan 2\phi}\right)\) [4]
2. You are given that the relevant solution of the equation given in part (c) is \(\phi = 1.0207\) correct to 5 significant figures. Find the distance from \(A\) to the pole. Give your answer correct to 3 significant figures. [1]

A curve has polar equation \(r = a \sin 3\theta\) for \(-\frac{1}{4}\pi \leq \theta \leq \frac{1}{4}\pi\), where \(a\) is a positive constant.

1. Sketch the curve. [2]
2. In this question you must show detailed reasoning. Find, in terms of \(a\) and \(\pi\), the area enclosed by one of the loops of the curve. [5]

1. Show that \(\tan\theta\) may be expressed as \(\frac{2t}{1-t^2}\), where \(t = \tan\left(\frac{\theta}{2}\right)\). [1]

The diagram below shows a sketch of the curve \(C\) with polar equation $$r = \cos\left(\frac{\theta}{2}\right), \quad \text{where } -\pi < \theta \leqslant \pi.$$ \includegraphics{figure_6}

1. Show that the \(\theta\)-coordinate of the points at which the tangent to \(C\) is perpendicular to the initial line satisfies the equation $$\tan\theta = -\frac{1}{2}\tan\left(\frac{\theta}{2}\right).$$ [4]
2. Hence, find the polar coordinates of the points on \(C\) where the tangent is perpendicular to the initial line. [6]
3. Calculate the area of the region enclosed by the curve \(C\) and the initial line for \(0 \leqslant \theta \leqslant \pi\). [5]

1. Given that \(z^n + \frac{1}{z^n} = 2\cos n\theta\), where \(z = \cos\theta + \mathrm{i}\sin\theta\), express \(16\cos^4\theta\) in the form $$a\cos 4\theta + b\cos 2\theta + c,$$ where \(a\), \(b\), \(c\) are integers whose values are to be determined. [5]

The diagram below shows a sketch of the curve C with polar equation $$r = 3 - 4\cos^2\theta, \quad \text{where } \frac{\pi}{6} \leq \theta \leq \frac{5\pi}{6}.$$ \includegraphics{figure_4}

1. Calculate the area of the region enclosed by the curve C. [8]
2. Find the exact polar coordinates of the points on C at which the tangent is perpendicular to the initial line. [8]