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

The polar equation of a curve \(C\) is \(r = a ( 1 + \cos \theta )\) for \(0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant.

1. Sketch \(C\).
2. Show that the cartesian equation of \(C\) is $$x ^ { 2 } + y ^ { 2 } = a \left( x + \sqrt { } \left( x ^ { 2 } + y ^ { 2 } \right) \right)$$
3. Find the area of the sector of \(C\) between \(\theta = 0\) and \(\theta = \frac { 1 } { 3 } \pi\).
4. Find the arc length of \(C\) between the point where \(\theta = 0\) and the point where \(\theta = \frac { 1 } { 3 } \pi\).

The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations, for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), as follows: $$\begin{aligned} & C _ { 1 } : r = 2 \left( \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } \right) , \\ & C _ { 2 } : r = \mathrm { e } ^ { 2 \theta } - \mathrm { e } ^ { - 2 \theta } \end{aligned}$$ The curves intersect at the point \(P\) where \(\theta = \alpha\).

1. Show that \(\mathrm { e } ^ { 2 \alpha } - 2 \mathrm { e } ^ { \alpha } - 1 = 0\). Hence find the exact value of \(\alpha\) and show that the value of \(r\) at \(P\) is \(4 \sqrt { } 2\).
2. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
3. Find the area of the region enclosed by \(C _ { 1 } , C _ { 2 }\) and the initial line, giving your answer correct to 3 significant figures.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

The curve \(C\) has polar equation \(r = a ( 1 - \cos \theta )\) for \(0 \leqslant \theta < 2 \pi\).

1. Sketch \(C\).
2. Find the area of the region enclosed by the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\), the half-line \(\theta = \frac { 1 } { 2 } \pi\) and the half-line \(\theta = \frac { 3 } { 2 } \pi\).
3. Show that $$\left( \frac { \mathrm { d } s } { \mathrm {~d} \theta } \right) ^ { 2 } = 4 a ^ { 2 } \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)$$ where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\).

7 The curve \(C\) has polar equation $$r = \theta \sin \theta ,$$ where \(0 \leqslant \theta \leqslant \pi\). Draw a sketch of \(C\). Find the area of the region enclosed by \(C\), leaving your answer in terms of \(\pi\).

10 The curve \(C\) has polar equation \(r = 3 + 2 \cos \theta\), for \(- \pi < \theta \leqslant \pi\). The straight line \(l\) has polar equation \(r \cos \theta = 2\). Sketch both \(C\) and \(l\) on a single diagram. Find the polar coordinates of the points of intersection of \(C\) and \(l\). The region \(R\) is enclosed by \(C\) and \(l\), and contains the pole. Find the area of \(R\).

5 The curve \(C\) has polar equation \(r = 1 + 2 \cos \theta\). Sketch the curve for \(- \frac { 2 } { 3 } \pi \leqslant \theta < \frac { 2 } { 3 } \pi\). Find the area bounded by \(C\) and the half-lines \(\theta = - \frac { 1 } { 3 } \pi , \theta = \frac { 1 } { 3 } \pi\).

1 The curve \(C\) has polar equation \(r = 2 \mathrm { e } ^ { \theta }\), for \(\frac { 1 } { 6 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Find

1. the area of the region bounded by the half-lines \(\theta = \frac { 1 } { 6 } \pi , \theta = \frac { 1 } { 2 } \pi\) and \(C\),
2. the length of \(C\).

2 The diagram shows a sketch of part of the curve \(C\) whose polar equation is \(r = 1 + \tan \theta\). The point \(O\) is the pole. \includegraphics[max width=\textwidth, alt={}, center]{0c177d90-02ae-4e91-bc9d-d0c7051799b8-3_561_629_406_772} The points \(P\) and \(Q\) on the curve are given by \(\theta = 0\) and \(\theta = \frac { \pi } { 3 }\) respectively.

1. Show that the area of the region bounded by the curve \(C\) and the lines \(O P\) and \(O Q\) is $$\frac { 1 } { 2 } \sqrt { 3 } + \ln 2$$ (6 marks)
2. Hence find the area of the shaded region bounded by the line \(P Q\) and the arc \(P Q\) of \(C\).

6 A curve \(C\) has polar equation $$r ^ { 2 } \sin 2 \theta = 8$$

1. Find the cartesian equation of \(C\) in the form \(y = \mathrm { f } ( x )\).
2. Sketch the curve \(C\).
3. The line with polar equation \(r = 2 \sec \theta\) intersects \(C\) at the point \(A\). Find the polar coordinates of \(A\).

5 The diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line. \includegraphics[max width=\textwidth, alt={}, center]{f4fdffc7-5647-4462-a983-1564d4e76a4d-3_301_668_1644_689} The curve \(C\) has polar equation $$r = \frac { 2 } { 3 + 2 \cos \theta } , \quad 0 \leqslant \theta \leqslant 2 \pi$$

1. Verify that the point \(L\) with polar coordinates ( \(2 , \pi\) ) lies on \(C\).
2. The circle with polar equation \(r = 1\) intersects \(C\) at the points \(M\) and \(N\).
   1. Find the polar coordinates of \(M\) and \(N\).
   2. Find the area of triangle \(L M N\).
3. Find a cartesian equation of \(C\), giving your answer in the form \(9 y ^ { 2 } = \mathrm { f } ( x )\).

3 A curve \(C\) has polar equation \(r ( 1 + \cos \theta ) = 2\).

1. Find the cartesian equation of \(C\), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
2. The straight line with polar equation \(4 r = 3 \sec \theta\) intersects the curve \(C\) at the points \(P\) and \(Q\). Find the length of \(P Q\).

3 The diagram shows a sketch of a circle which passes through the origin \(O\). \includegraphics[max width=\textwidth, alt={}, center]{13cfb9ca-9495-4b69-80c5-9fb7e8e0f957-3_423_451_356_794} The equation of the circle is \(( x - 3 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 25\) and \(O A\) is a diameter.

1. Find the cartesian coordinates of the point \(A\).
2. Using \(O\) as the pole and the positive \(x\)-axis as the initial line, the polar coordinates of \(A\) are \(( k , \alpha )\).
   1. Find the value of \(k\) and the value of \(\tan \alpha\).
   2. Find the polar equation of the circle \(( x - 3 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 25\), giving your answer in the form \(r = p \cos \theta + q \sin \theta\).

7 The diagram shows the curve \(C _ { 1 }\) with polar equation $$r = 1 + 6 \mathrm { e } ^ { - \frac { \theta } { \pi } } , \quad 0 \leqslant \theta \leqslant 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{13cfb9ca-9495-4b69-80c5-9fb7e8e0f957-4_300_513_1414_760}

1. Find, in terms of \(\pi\) and e , the area of the shaded region bounded by \(C _ { 1 }\) and the initial line.
2. The polar equation of a curve \(C _ { 2 }\) is $$r = \mathrm { e } ^ { \frac { \theta } { \pi } } , \quad 0 \leqslant \theta \leqslant 2 \pi$$ Sketch the curve \(C _ { 2 }\) and state the polar coordinates of the end-points of this curve.
3. The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the point \(P\). Find the polar coordinates of \(P\).

6 The polar equation of a curve \(C _ { 1 }\) is $$r = 2 ( \cos \theta - \sin \theta ) , \quad 0 \leqslant \theta \leqslant 2 \pi$$

1. 1. Find the cartesian equation of \(C _ { 1 }\).
   2. Deduce that \(C _ { 1 }\) is a circle and find its radius and the cartesian coordinates of its centre.
2. The diagram shows the curve \(C _ { 2 }\) with polar equation $$r = 4 + \sin \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{90a59b47-3799-46a2-b76b-ced5cc3e1aac-4_519_847_443_593}
   1. Find the area of the region that is bounded by \(C _ { 2 }\).
   2. Prove that the curves \(C _ { 1 }\) and \(C _ { 2 }\) do not intersect.
   3. Find the area of the region that is outside \(C _ { 1 }\) but inside \(C _ { 2 }\).

6 A differential equation is given by $$\left( x ^ { 3 } + 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } - 3 x ^ { 2 } \frac { d y } { d x } = 2 - 4 x ^ { 3 }$$

1. Show that the substitution $$u = \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 x$$ transforms this differential equation into $$\left( x ^ { 3 } + 1 \right) \frac { \mathrm { d } u } { \mathrm {~d} x } = 3 x ^ { 2 } u$$ (4 marks)
2. Hence find the general solution of the differential equation $$\left( x ^ { 3 } + 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 3 x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = 2 - 4 x ^ { 3 }$$ giving your answer in the form \(y = \mathrm { f } ( x )\). \(7 \quad\) The curve \(C _ { 1 }\) is defined by \(r = 2 \sin \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }\). The curve \(C _ { 2 }\) is defined by \(r = \tan \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }\).
   1. Find a cartesian equation of \(C _ { 1 }\).
   2. 1. Prove that the curves \(C _ { 1 }\) and \(C _ { 2 }\) meet at the pole \(O\) and at one other point, \(P\), in the given domain. State the polar coordinates of \(P\).
      2. The point \(A\) is the point on the curve \(C _ { 1 }\) at which \(\theta = \frac { \pi } { 4 }\). The point \(B\) is the point on the curve \(C _ { 2 }\) at which \(\theta = \frac { \pi } { 4 }\). Determine which of the points \(A\) or \(B\) is further away from the pole \(O\), justifying your answer.
      3. Show that the area of the region bounded by the arc \(O P\) of \(C _ { 1 }\) and the arc \(O P\) of \(C _ { 2 }\) is \(a \pi + b \sqrt { 3 }\), where \(a\) and \(b\) are rational numbers.

8

1. A curve has cartesian equation \(x y = 8\). Show that the polar equation of the curve is \(r ^ { 2 } = 16 \operatorname { cosec } 2 \theta\).
2. The diagram shows a sketch of the curve, \(C\), whose polar equation is $$r ^ { 2 } = 16 \operatorname { cosec } 2 \theta , \quad 0 < \theta < \frac { \pi } { 2 }$$ \includegraphics[max width=\textwidth, alt={}, center]{c4bce668-61f1-4be0-97ee-c635df7e1fc6-4_364_567_1635_726}
   1. Find the polar coordinates of the point \(N\) which lies on the curve \(C\) and is closest to the pole \(O\).
   2. The circle whose polar equation is \(r = 4 \sqrt { 2 }\) intersects the curve \(C\) at the points \(P\) and \(Q\). Find, in an exact form, the polar coordinates of \(P\) and \(Q\).
   3. The obtuse angle \(P N Q\) is \(\alpha\) radians. Find the value of \(\alpha\), giving your answer to three significant figures.
      (5 marks)

8 The diagram shows a sketch of a curve and a circle. \includegraphics[max width=\textwidth, alt={}, center]{a2bc95fe-5588-4ff7-a8a3-0cd07df412c9-4_460_693_370_680} The polar equation of the curve is $$r = 3 + 2 \sin \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$ The circle, whose polar equation is \(r = 2\), intersects the curve at the points \(P\) and \(Q\), as shown in the diagram.

1. Find the polar coordinates of \(P\) and the polar coordinates of \(Q\).
2. A straight line, drawn from the point \(P\) through the pole \(O\), intersects the curve again at the point \(A\).
   1. Find the polar coordinates of \(A\).
   2. Find, in surd form, the length of \(A Q\).
   3. Hence, or otherwise, explain why the line \(A Q\) is a tangent to the circle \(r = 2\).
3. Find the area of the shaded region which lies inside the circle \(r = 2\) but outside the curve \(r = 3 + 2 \sin \theta\). Give your answer in the form \(\frac { 1 } { 6 } ( m \sqrt { 3 } + n \pi )\), where \(m\) and \(n\) are integers.

4

1. The curve with Cartesian equation \(\frac { x ^ { 2 } } { c } + \frac { y ^ { 2 } } { d } = 1\) is mapped onto the curve with polar equation \(r = \frac { 10 } { 3 - 2 \cos \theta }\) by a single geometrical transformation. By writing the polar equation as a Cartesian equation in a suitable form, find the values of the constants \(c\) and \(d\).
2. Hence describe the geometrical transformation referred to in part (a).
   [0pt] [1 mark]

8 The diagram shows the sketch of part of a curve, the pole \(O\) and the initial line. \includegraphics[max width=\textwidth, alt={}, center]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-20_609_670_358_703} The polar equation of the curve is \(r = 1 + \tan \theta\).
The point \(A\) is the point on the curve at which \(\theta = \frac { \pi } { 3 }\).
The perpendicular, \(A N\), from \(A\) to the initial line intersects the curve at the point \(B\).

1. Find the exact length of \(O A\).
2. 1. Given that, at the point \(B , \theta = \alpha\), show that \(( \cos \alpha + \sin \alpha ) ^ { 2 } = 1 + \frac { \sqrt { 3 } } { 2 }\).
   2. Hence, or otherwise, find \(\alpha\) in terms of \(\pi\).
3. Show that the area of triangle \(O A B\) is \(\frac { 3 + 2 \sqrt { 3 } } { 6 }\).
4. Find, in an exact simplified form, the area of the shaded region bounded by the curve and the line segment \(A B\).
   [0pt] [7 marks]
   \includegraphics[max width=\textwidth, alt={}]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-23_2488_1709_219_153}
   \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}

5 The diagram below shows the curve \(C\) with polar equation \(r = 3 ( 1 - \sin 2 \theta )\) for \(0 \leqslant \theta \leqslant 2 \pi\). \includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-5_728_963_303_239}

1. Show that a cartesian equation of \(C\) is \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = 9 ( x - y ) ^ { 4 }\).
2. Show that the line with equation \(\mathrm { y } = \mathrm { x }\) is a line of symmetry of \(C\).
3. In this question you must show detailed reasoning. Find the exact area of each of the loops of \(C\).

11 A curve has cartesian equation \(x ^ { 3 } + y ^ { 3 } = 2 x y\). \(C\) is the portion of the curve for which \(x \geqslant 0\) and \(y \geqslant 0\). The equation of \(C\) in polar form is given by \(r = \mathrm { f } ( \theta )\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Find \(f ( \theta )\).
2. Find an expression for \(\mathrm { f } \left( \frac { 1 } { 2 } \pi - \theta \right)\), giving your answer in terms of \(\sin \theta\) and \(\cos \theta\).
3. Hence find the line of symmetry of \(C\).
4. Find the value of \(r\) when \(\theta = \frac { 1 } { 4 } \pi\).
5. By finding values of \(\theta\) when \(r = 0\), show that \(C\) has a loop.

2 Two polar curves, \(C _ { 1 }\) and \(C _ { 2 }\), are defined by \(C _ { 1 } : r = 2 \theta\) and \(C _ { 2 } : r = \theta + 1\) where \(0 \leqslant \theta \leqslant 2 \pi\). \(C _ { 1 }\) intersects the initial line at two points, the pole and the point \(A\).

1. Write down the polar coordinates of \(A\).
2. Determine the polar coordinates of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\). The diagram below shows a sketch of \(C _ { 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{007f07ee-cb29-4a97-93d9-2328079c4aea-2_681_1353_1318_244}
3. On the copy of this sketch in the Printed Answer Booklet, sketch \(C _ { 2 }\).

6 The equation of a curve in polar coordinates is \(r = \ln ( 1 + \sin \theta )\) for \(\alpha \leqslant \theta \leqslant \beta\) where \(\alpha\) and \(\beta\) are non-negative angles. The curve consists of a single closed loop through the pole.

1. By solving the equation \(r = 0\), determine the smallest possible values of \(\alpha\) and \(\beta\).
2. Find the area enclosed by the curve, giving your answer to 4 significant figures.
3. Hence, by considering the value of \(r\) at \(\theta = \frac { \alpha + \beta } { 2 }\), show that the loop is not circular.

7 A curve has cartesian equation \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 c ^ { 2 } x y\), where \(c\) is a positive constant.

1. Show that the polar equation of the curve is \(r ^ { 2 } = c ^ { 2 } \sin 2 \theta\).
2. Sketch the curves \(r = c \sqrt { \sin 2 \theta }\) and \(r = - c \sqrt { \sin 2 \theta }\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
3. Find the area of the region enclosed by one of the loops in part (b). Section B (110 marks)
   Answer all the questions.

5

1. Sketch the polar curve \(\mathrm { r } = \mathrm { a } ( 1 - \cos \theta ) , 0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant.
2. Determine the exact area of the region enclosed by the curve.