4.09c Area enclosed: by polar curve

3 The diagram shows a sketch of a loop, the pole \(O\) and the initial line. \includegraphics[max width=\textwidth, alt={}, center]{f4fdffc7-5647-4462-a983-1564d4e76a4d-3_305_553_383_740} The polar equation of the loop is $$r = ( 2 + \cos \theta ) \sqrt { \sin \theta } , \quad 0 \leqslant \theta \leqslant \pi$$ Find the area enclosed by the loop.

8 The diagram shows a sketch of a curve \(C\) and a line \(L\), which is parallel to the initial line and touches the curve at the points \(P\) and \(Q\). \includegraphics[max width=\textwidth, alt={}, center]{32de7ef6-b7aa-4bfd-a73a-e12bfc0147e2-5_506_762_447_639} The polar equation of the curve \(C\) is $$r = 4 ( 1 - \sin \theta ) , \quad 0 \leqslant \theta < 2 \pi$$ and the polar equation of the line \(L\) is $$r \sin \theta = 1$$

1. Show that the polar coordinates of \(P\) are \(\left( 2 , \frac { \pi } { 6 } \right)\) and find the polar coordinates of \(Q\).
2. Find the area of the shaded region \(R\) bounded by the line \(L\) and the curve \(C\). Give your answer in the form \(m \sqrt { 3 } + n \pi\), where \(m\) and \(n\) are integers.

6 The diagram shows a sketch of a curve \(C\). \includegraphics[max width=\textwidth, alt={}, center]{8cb4b110-274e-47ec-a31b-ee8f84434a65-3_305_556_1078_721} The polar equation of the curve is $$r = 2 \sin 2 \theta \sqrt { \cos \theta } , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ Show that the area of the region bounded by \(C\) is \(\frac { 16 } { 15 }\).

8 The diagram shows a sketch of the curve \(C\) with polar equation $$r = 3 + 2 \cos \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{80c4336c-b0ca-46de-a871-812e6923f7f2-4_463_668_1468_699}

1. Find the area of the region bounded by the curve \(C\).
2. A circle, whose cartesian equation is \(( x - 4 ) ^ { 2 } + y ^ { 2 } = 16\), intersects the curve \(C\) at the points \(A\) and \(B\).
   1. Find, in surd form, the length of \(A B\).
   2. Find the perimeter of the segment \(A O B\) of the circle, where \(O\) is the pole.

8 The diagram shows a sketch of a curve. \includegraphics[max width=\textwidth, alt={}, center]{f05737eb-adb1-4228-aebf-6b5c7f26a434-5_464_574_402_726} The polar equation of the curve is $$r = \sin 2 \theta \sqrt { \left( 2 + \frac { 1 } { 2 } \cos \theta \right) } , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ The point \(P\) is the point of the curve at which \(\theta = \frac { \pi } { 3 }\). The perpendicular from \(P\) to the initial line meets the initial line at the point \(N\).

1. 1. Find the exact value of \(r\) when \(\theta = \frac { \pi } { 3 }\).
   2. Show that the polar equation of the line \(P N\) is \(r = \frac { 3 \sqrt { 3 } } { 8 } \sec \theta\).
   3. Find the area of triangle \(O N P\) in the form \(\frac { k \sqrt { 3 } } { 128 }\), where \(k\) is an integer.
2. 1. Using the substitution \(u = \sin \theta\), or otherwise, find \(\int \sin ^ { n } \theta \cos \theta \mathrm {~d} \theta\), where \(n \geqslant 2\).
   2. Find the area of the shaded region bounded by the line \(O P\) and the arc \(O P\) of the curve. Give your answer in the form \(a \pi + b \sqrt { 3 } + c\), where \(a , b\) and \(c\) are constants.
      (8 marks)

4 The diagram shows the curve \(C\) with polar equation $$r = 6 ( 1 - \cos \theta ) , \quad 0 \leqslant \theta < 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{06ae13de-5cf3-421d-ac7a-ee9f74b653be-3_552_903_922_550}

1. Find the area of the region bounded by the curve \(C\).
2. The circle with cartesian equation \(x ^ { 2 } + y ^ { 2 } = 9\) intersects the curve \(C\) at the points \(A\) and \(B\).
   1. Find the polar coordinates of \(A\) and \(B\).
   2. Find, in surd form, the length of \(A B\).

8 The polar equation of a curve \(C\) is $$r = 5 + 2 \cos \theta , \quad - \pi \leqslant \theta \leqslant \pi$$

1. Verify that the points \(A\) and \(B\), with polar coordinates ( 7,0 ) and ( \(3 , \pi\) ) respectively, lie on the curve \(C\).
2. Sketch the curve \(C\).
3. Find the area of the region bounded by the curve \(C\).
4. The point \(P\) is the point on the curve \(C\) for which \(\theta = \alpha\), where \(0 < \alpha \leqslant \frac { \pi } { 2 }\). The point \(Q\) lies on the curve such that \(P O Q\) is a straight line, where the point \(O\) is the pole. Find, in terms of \(\alpha\), the area of triangle \(O Q B\).

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 \(C _ { 2 }\).

3 The diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line. \includegraphics[max width=\textwidth, alt={}, center]{c4bce668-61f1-4be0-97ee-c635df7e1fc6-2_380_735_1827_648} The polar equation of \(C\) is $$r = 2 \sqrt { 1 + \tan \theta } , \quad - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 }$$ Show that the area of the shaded region, bounded by the curve \(C\) and the initial line, is \(\frac { \pi } { 2 } - \ln 2\).
(4 marks)

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.

8 The diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line. The curve \(C\) intersects the initial line at the point \(P\). \includegraphics[max width=\textwidth, alt={}, center]{0eb3e96e-528c-4a99-b164-31cc865f0d68-20_432_949_402_525} The polar equation of \(C\) is \(r = \left( 1 - \tan ^ { 2 } \theta \right) \sec \theta , - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 }\).

1. Show that the area of the region bounded by the curve \(C\) is \(\frac { 8 } { 15 }\).
2. The curve whose polar equation is $$r = \frac { 1 } { 2 } \sec ^ { 3 } \theta , \quad - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 }$$ intersects \(C\) at the points \(A\) and \(B\).
   1. Find the polar coordinates of \(A\) and \(B\).
   2. Given that angle \(O A P =\) angle \(O B P = \alpha\), show that \(\tan \alpha = k \sqrt { 3 }\), where \(k\) is an integer.
   3. Using your value of \(k\) from part (b)(ii), state whether the point \(A\) lies inside or lies outside the circle whose diameter is \(O P\). Give a reason for your answer.
      [0pt] [1 mark]
      \includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-21_2484_1707_221_153}
      \includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-23_2484_1707_221_153}

7 The diagram shows the sketch of a curve \(C _ { 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-18_362_734_360_635} The polar equation of the curve \(C _ { 1 }\) is $$r = 1 + \cos 2 \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$

1. Find the area of the region bounded by the curve \(C _ { 1 }\).
2. The curve \(C _ { 2 }\) whose polar equation is $$r = 1 + \sin \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$ intersects the curve \(C _ { 1 }\) at the pole \(O\) and at the point \(A\). The straight line drawn through \(A\) parallel to the initial line intersects \(C _ { 1 }\) again at the point \(B\).
   1. Find the polar coordinates of \(A\).
   2. Show that the length of \(O B\) is \(\frac { 1 } { 4 } ( \sqrt { 13 } + 1 )\).
   3. Find the length of \(A B\), giving your answer to three significant figures. \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-22_2486_1728_221_141} \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-23_2486_1728_221_141} \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-24_2488_1728_219_141}

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 \(C\).

12 For any positive parameter \(k\), the curve \(C _ { k }\) is defined by the polar equation \(\mathrm { r } = \mathrm { k } ( \cos \theta + 1 ) + \frac { 10 } { \mathrm { k } } , 0 \leqslant \theta \leqslant 2 \pi\).
For each value of \(k\) the curve is a single, closed loop with no self-intersections. The diagram shows \(C _ { 10.5 }\) for the purpose of illustration. \includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-6_558_723_550_242} Each curve, \(C _ { k }\), encloses a certain area, \(A _ { k }\).
You are given that there is a single minimum value of \(A _ { k }\).
Determine, in an exact form, the value of \(k\) for which \(C _ { k }\) encloses this minimum area.

3 In this question you must show detailed reasoning. The diagram below shows the curve \(r = 2 \cos 4 \theta\) for \(- k \pi \leq \theta \leq k \pi\) where \(k\) is a constant to be determined. Calculate the exact area enclosed by the curve.

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.

9 A curve has polar equation \(r = \operatorname { asin } 3 \theta\), for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant.

1. Sketch the curve. Indicate the parts of the curve where \(r\) is negative by using a broken line.
2. In this question you must show detailed reasoning. Determine the area of one of the loops of the curve.

5 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. \begin{figure}[h] \end{figure}

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

1 A family of curves is given by the parametric equations \(x ( t ) = \cos ( t ) - \frac { \cos ( ( m + 1 ) t ) } { m + 1 }\) and \(y ( t ) = \sin ( t ) - \frac { \sin ( ( m + 1 ) t ) } { m + 1 }\) where \(0 \leqslant t < 2 \pi\) and \(m\) is a positive integer.

1. 1. Sketch the curves in the cases \(m = 3 , m = 4\) and \(m = 5\) on separate axes in the Printed Answer Booklet.
   2. State one common feature of these three curves.
   3. State a feature for the case \(m = 4\) which is absent in the cases \(m = 3\) and \(m = 5\).
2. 1. Determine, in terms of \(m\), the values of \(t\) for which \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\) but \(\frac { \mathrm { d } y } { \mathrm {~d} t } \neq 0\).
   2. Describe the tangent to the curve at the points corresponding to such values of \(t\).
3. 1. Show that the curve lies between the circle centred at the origin with radius $$1 - \frac { 1 } { m + 1 }$$ and the circle centred at the origin with radius $$1 + \frac { 1 } { m + 1 }$$
   2. Hence, or otherwise, show that the area \(A\) bounded by the curve satisfies $$\frac { m ^ { 2 } \pi } { ( m + 1 ) ^ { 2 } } < A < \frac { ( m + 2 ) ^ { 2 } \pi } { ( m + 1 ) ^ { 2 } }$$
   3. Find the limit of the area bounded by the curve as \(m\) tends to infinity.
4. The arc length of a curve defined by parametric equations \(x ( t )\) and \(y ( t )\) between points corresponding to \(t = c\) and \(t = d\), where \(c < d\), is $$\int _ { c } ^ { d } \sqrt { \left( \frac { \mathrm {~d} x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } } \mathrm {~d} t$$ Use this to show that the length of the curve is independent of \(m\).

1

1. A family of curves is given by the equation $$x ^ { 2 } + y ^ { 2 } + 2 a x y = 1 ( * )$$ where the parameter \(a\) is a real number.
   You may find it helpful to use a slider (for \(a\) ) to investigate this family of curves.
   1. On the axes in the Printed Answer Booklet, sketch the curve in each of the cases
      • \(a = 0\)
      • \(a = 0.5\)
      • \(a = 2\)
      • State a feature of the curve for the cases \(a = 0 , a = 0.5\) that is not a feature of the curve in the case \(a = 2\).
      • In the case \(a = 1\), the curve consists of two straight lines. Determine the equations of these lines.
      • 1. Find an equation of the curve (*) in polar form.
        2. Hence, or otherwise, find, in exact form, the area bounded by the curve, the positive part of the \(x\)-axis and the positive part of the \(y\)-axis, in the case \(a = 2\).
2. In this part of the question \(m\) is any real number.
Describing all possible cases, determine the pairs of values \(a\) and \(m\) for which the curve with equation (*) intersects the straight line given by \(y = m x\).