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

208 questions

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Edexcel FP2 Q28
16 marks Challenging +1.8
  1. Sketch the curve with polar equation $$r = 3 \cos 2\theta, \quad -\frac{\pi}{4} \leq \theta < \frac{\pi}{4}.$$ [2]
  2. Find the area of the smaller finite region enclosed between the curve and the half-line \(\theta = \frac{\pi}{6}\). [6]
  3. Find the exact distance between the two tangents which are parallel to the initial line. [8]
Edexcel FP2 Q32
16 marks Challenging +1.2
\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]
Edexcel FP2 Q40
13 marks Standard +0.8
The curve \(C\) has polar equation \(r = 6 \cos \theta\), \(-\frac{\pi}{2} \leq \theta < \frac{\pi}{2}\), and the line \(D\) has polar equation \(r = 3 \sec\left(\frac{\pi}{3} - \theta\right)\), \(-\frac{\pi}{6} \leq \theta \leq \frac{5\pi}{6}\).
  1. Find a cartesian equation of \(C\) and a cartesian equation of \(D\). [5]
  2. Sketch on the same diagram the graphs of \(C\) and \(D\), indicating where each cuts the initial line. [3] The graphs of \(C\) and \(D\) intersect at the points \(P\) and \(Q\).
  3. Find the polar coordinates of \(P\) and \(Q\). [5]
Edexcel FP2 Q45
13 marks Challenging +1.3
\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]
Edexcel FP3 Q1
5 marks Moderate -0.8
An ellipse has equation \(\frac{x^2}{16} + \frac{y^2}{9} = 1\).
  1. Sketch the ellipse. [1]
  2. Find the value of the eccentricity \(e\). [2]
  3. State the coordinates of the foci of the ellipse. [2]
Edexcel FP3 Q39
7 marks Standard +0.3
The hyperbola \(H\) has equation \(\frac{x^2}{16} - \frac{y^2}{4} = 1\). Find
  1. the value of the eccentricity of \(H\), [2]
  2. the distance between the foci of \(H\). [2]
The ellipse \(E\) has equation \(\frac{x^2}{16} + \frac{y^2}{4} = 1\).
  1. Sketch \(H\) and \(E\) on the same diagram, showing the coordinates of the points where each curve crosses the axes. [3]
OCR FP2 2010 January Q4
7 marks Standard +0.8
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]
AQA Further AS Paper 1 2018 June Q4
2 marks Standard +0.3
Sketch the graph given by the polar equation $$r = \frac{a}{\cos \theta}$$ where \(a\) is a positive constant. [2 marks] \includegraphics{figure_4}
AQA Further AS Paper 1 2019 June Q4
2 marks Moderate -0.3
The line \(L\) has polar equation $$r = \frac{k}{\sin \theta}$$ where \(k\) is a positive constant.
  1. Sketch \(L\). [1 mark]
  2. State the minimum distance between \(L\) and the point \(O\). [1 mark]
AQA Further AS Paper 1 2020 June Q11
3 marks Challenging +1.2
Sketch the polar graph of $$r = \sinh \theta + \cosh \theta$$ for \(0 \leq \theta \leq 2\pi\) \includegraphics{figure_11} [3 marks]
AQA Further Paper 1 2019 June Q15
11 marks Challenging +1.8
The diagram shows part of a spiral curve. The point \(P\) has polar coordinates \((r, \theta)\) where \(0 \leq \theta \leq \frac{\pi}{2}\) The points \(T\) and \(S\) lie on the initial line and \(O\) is the pole. \(TPQ\) is the tangent to the curve at \(P\). \includegraphics{figure_15}
  1. Show that the gradient of \(TPQ\) is equal to $$\frac{\frac{dr}{d\theta} \sin \theta + r \cos \theta}{\frac{dr}{d\theta} \cos \theta - r \sin \theta}$$ [4 marks]
  2. The curve has polar equation $$r = e^{(\cot b)\theta}$$ where \(b\) is a constant such that \(0 < b < \frac{\pi}{2}\) Use the result of part (a) to show that the angle between the line \(OP\) and the tangent \(TPQ\) does not depend on \(\theta\). [7 marks]
AQA Further Paper 1 2022 June Q9
14 marks Challenging +1.8
Roberto is solving this mathematics problem:
The curve \(C_1\) has polar equation
\(r^2 = 9\sin 2\theta\)
for all possible values of \(\theta\)
Find the area enclosed by \(C_1\)
Roberto's solution is as follows:
\(A = \frac{1}{2}\int_{-\pi}^{\pi} 9\sin 2\theta \, d\theta\)
\(= \left[-\frac{9}{4}\cos 2\theta\right]_{-\pi}^{\pi}\)
\(= 0\)
  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]
AQA Further Paper 1 2023 June Q14
10 marks Challenging +1.2
The curve C has polar equation $$r = \frac{A}{5 + 3 \cos \theta} \quad (-\pi < \theta \leq \pi)$$
  1. Show that \(r\) takes values in the range \(\frac{1}{k} \leq r \leq k\), where \(k\) is an integer. [2 marks]
  2. Find the Cartesian equation of C in the form \(y^2 = f(x)\) [4 marks]
  3. The ellipse E has equation $$y^2 + \frac{16x^2}{25} = 1$$ Find the transformation that maps the graph of E onto C [4 marks]
AQA Further Paper 1 2024 June Q16
9 marks Challenging +1.8
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]
AQA Further Paper 2 2020 June Q14
11 marks Hard +2.3
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]
AQA Further Paper 2 Specimen Q11
8 marks Challenging +1.8
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]
OCR Further Pure Core 1 2021 November Q7
9 marks Standard +0.8
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]
OCR Further Pure Core 2 2024 June Q6
11 marks Challenging +1.8
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\(\frac{1}{12}\pi\)\(\frac{1}{6}\pi\)\(\frac{1}{4}\pi\)\(\frac{1}{3}\pi\)\(\frac{5}{12}\pi\)\(\frac{1}{2}\pi\)
\(r\)00.2621.851
  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]
OCR Further Pure Core 2 Specimen Q9
6 marks Standard +0.8
A curve has equation \(x^4 + y^4 = x^2 + y^2\), where \(x\) and \(y\) are not both zero. \begin{enumerate}[label=(\roman*)] \item Show that the equation of the curve in polar coordinates is \(r^2 = \frac{2}{2-\sin^2 2\theta}\). [4] \item Deduce that no point on the curve \(x^4 + y^4 = x^2 + y^2\) is further than \(\sqrt{2}\) from the origin. [2]
OCR MEI Further Pure Core Specimen Q9
7 marks Challenging +1.3
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]
WJEC Further Unit 4 2019 June Q8
10 marks Challenging +1.8
The curve \(C\) has polar equation $$r = \sin 2\theta, \quad \text{where} \quad 0 < \theta \leqslant \frac{\pi}{2}.$$
  1. Find the polar coordinates of the point on \(C\) at which the tangent is parallel to the initial line. Give your answers correct to three decimal places. [9]
  2. Write the coordinates of this point in Cartesian form. [1]
WJEC Further Unit 4 2022 June Q13
11 marks Challenging +1.3
The curve C has polar equation \(r = 2 - \cos\theta\) for \(0 \leq \theta \leq 2\pi\).
  1. Sketch the curve C. [2]
    1. Show that the values of \(\theta\) at which the tangent to the curve \(r = 2 - \cos\theta\) is parallel to the initial line satisfy the equation $$2\cos^2\theta - 2\cos\theta - 1 = 0.$$
    2. Find the polar coordinates of the points where the tangent to the curve \(r = 2 - \cos\theta\) is parallel to the initial line. [9]
WJEC Further Unit 4 2023 June Q6
16 marks Challenging +1.8
  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]
SPS SPS FM Pure 2021 May Q2
9 marks Standard +0.3
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_2}
  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]
SPS SPS FM Pure 2022 February Q6
13 marks Challenging +1.8
The curve \(C\) has equation $$r = a(p + 2\cos\theta) \quad 0 \leqslant \theta < 2\pi$$ where \(a\) and \(p\) are positive constants and \(p > 2\) There are exactly four points on \(C\) where the tangent is perpendicular to the initial line.
  1. Show that the range of possible values for \(p\) is $$2 < p < 4$$ [5]
  2. Sketch the curve with equation $$r = a(3 + 2\cos\theta) \quad 0 \leqslant \theta < 2\pi \quad \text{where } a > 0$$ [1]
John digs a hole in his garden in order to make a pond. The pond has a uniform horizontal cross section that is modelled by the curve with equation $$r = 20(3 + 2\cos\theta) \quad 0 \leqslant \theta < 2\pi$$ where \(r\) is measured in centimetres. The depth of the pond is 90 centimetres. Water flows through a hosepipe into the pond at a rate of 50 litres per minute. Given that the pond is initially empty,
  1. determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute. [7]