4.09a Polar coordinates: convert to/from cartesian

169 questions

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Edexcel FP2 Q16
16 marks Challenging +1.2
\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]
Edexcel FP2 Q20
14 marks Challenging +1.2
\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]
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 2009 January Q7
8 marks Challenging +1.3
\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]
OCR MEI FP2 2011 January Q1
19 marks Standard +0.3
  1. A curve has polar equation \(r = 2(\cos \theta + \sin \theta)\) for \(-\frac{1}{4}\pi \leq \theta \leq \frac{3}{4}\pi\).
    1. Show that a cartesian equation of the curve is \(x^2 + y^2 = 2x + 2y\). Hence or otherwise sketch the curve. [5]
    2. Find, by integration, the area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac{1}{2}\pi\). Give your answer in terms of \(\pi\). [7]
    1. Given that \(f(x) = \arctan(\frac{1}{2}x)\), find \(f'(x)\). [2]
    2. Expand \(f'(x)\) in ascending powers of \(x\) as far as the term in \(x^4\). Hence obtain an expression for \(f(x)\) in ascending powers of \(x\) as far as the term in \(x^5\). [5]
AQA Further AS Paper 1 2019 June Q3
1 marks Easy -1.8
Point \(P\) has polar coordinates \(\left(2, \frac{2\pi}{3}\right)\). Which of the following are the Cartesian coordinates of \(P\)? Circle your answer. [1 mark] \((1, -\sqrt{3})\) \quad \((-\sqrt{3}, 1)\) \quad \((\sqrt{3}, -1)\) \quad \((-1, \sqrt{3})\)
AQA Further AS Paper 1 2020 June Q17
4 marks Standard +0.8
The polar equation of the circle \(C\) is $$r = a(\cos \theta + \sin \theta)$$ Find, in terms of \(a\), the radius of \(C\). Fully justify your answer. [4 marks]
AQA Further Paper 1 2021 June Q3
1 marks Moderate -0.5
The curve C has polar equation $$r^2 \sin 2\theta = 4$$ Find a Cartesian equation for C. Circle your answer. [1 mark] \(y = 2x\) \quad \(y = \frac{x}{2}\) \quad \(y = \frac{2}{x}\) \quad \(y = 4x\)
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]
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]
SPS SPS FM 2021 November Q2
3 marks Standard +0.3
The equation of a curve in polar coordinates is $$r = 11 + 9 \sec \theta.$$ Show that a cartesian equation of the curve is $$(x - 9)\sqrt{x^2 + y^2} = 11x.$$ [3 marks]
SPS SPS FM Pure 2023 November Q2
8 marks Standard +0.3
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]
SPS SPS FM Pure 2023 November Q7
Challenging +1.8
    1. Show on an Argand diagram the locus of points given by the values of \(z\) satisfying $$|z - 4 - 3i| = 5$$ Taking the initial line as the positive real axis with the pole at the origin and given that $$\theta \in [\alpha, \alpha + \pi], \text{ where } \alpha = -\arctan\left(\frac{4}{3}\right),$$
    2. show that this locus of points can be represented by the polar curve with equation $$r = 8\cos\theta + 6\sin\theta$$ (6) The set of points \(A\) is defined by $$A = \left\{z : 0 \leqslant \arg z \leqslant \frac{\pi}{3}\right\} \cap \{z : |z - 4 - 3i| \leqslant 5\}$$
    1. Show, by shading on your Argand diagram, the set of points \(A\).
    2. Find the exact area of the region defined by \(A\), giving your answer in simplest form. (7)
SPS SPS FM Pure 2026 November Q3
9 marks Challenging +1.2
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]
Pre-U Pre-U 9795/1 2011 June Q13
18 marks Challenging +1.8
    1. Given that \(t = \tan x\), prove that \(\frac{2}{2 - \sin 2x} = \frac{1 + t^2}{1 - t + t^2}\). [2]
    2. Hence determine the value of the constant \(k\) for which $$\frac{d}{dx}\left\{\tan^{-1}\left(\frac{1 - 2\tan x}{\sqrt{3}}\right)\right\} = \frac{k}{2 - \sin 2x}.$$ [4]
  2. The curve \(C\) has cartesian equation \(x^2 - xy + y^2 = 72\).
    1. Determine a polar equation for \(C\) in the form \(r^2 = f(\theta)\), and deduce the polar coordinates \((r, \theta)\), where \(0 \leqslant \theta < 2\pi\), of the points on \(C\) which are furthest from the pole \(O\). [7]
    2. Find the exact area of the region of the plane in the first quadrant bounded by \(C\), the \(x\)-axis and the line \(y = x\). Deduce the total area of the region of the plane which lies inside \(C\) and within the first quadrant. [5]