Sketch polar curve

7 The curve \(C\) has polar equation \(r ^ { 2 } = \sin 2 \theta \cos \theta\), for \(0 \leqslant \theta \leqslant \pi\).

1. Sketch \(C\) and state the equation of the line of symmetry.
2. Find a Cartesian equation for \(C\).
   
3. Find the total area enclosed by \(C\).
4. Find the greatest distance of a point on \(C\) from the pole. \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-16_2718_36_141_2011} If you use the following page to complete the answer to any question, the question number must be clearly shown.
   

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6 The curve \(C\) has polar equation \(r = 2 \cos \theta ( 1 + \sin \theta )\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Find the polar coordinates of the point on \(C\) that is furthest from the pole.
2. Sketch C.
3. Find the area of the region bounded by \(C\) and the initial line, giving your answer in exact form.

5 The curve \(C\) has polar equation \(r = \operatorname { asec } ^ { 2 } \theta\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).

1. Sketch \(C\), stating the polar coordinates of the point of intersection of \(C\) with the initial line and also with the half-line \(\theta = \frac { 1 } { 4 } \pi\).
2. Find the maximum distance of a point of \(C\) from the initial line.
3. Find the area of the region enclosed by \(C\), the initial line and the half-line \(\theta = \frac { 1 } { 4 } \pi\).
4. Find, in the form \(y = f ( x )\), the Cartesian equation of \(C\).

1

1. A curve has cartesian equation \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 3 x y ^ { 2 }\).
   1. Show that the polar equation of the curve is \(r = 3 \cos \theta \sin ^ { 2 } \theta\).
   2. Hence sketch the curve.
2. Find the exact value of \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - 3 x ^ { 2 } } } \mathrm {~d} x\).
3. 1. Write down the series for \(\ln ( 1 + x )\) and the series for \(\ln ( 1 - x )\), both as far as the term in \(x ^ { 5 }\).
   2. Hence find the first three non-zero terms in the series for \(\ln \left( \frac { 1 + x } { 1 - x } \right)\).
   3. Use the series in part (ii) to show that \(\sum _ { r = 0 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) 4 ^ { r } } = \ln 3\).

8 The equation of a curve, in polar coordinates, is $$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$

1. \includegraphics[max width=\textwidth, alt={}, center]{63a316f6-1c18-4224-930f-0b58112c9f71-3_268_796_1567_717} The diagram shows the part of the curve for which \(0 \leqslant \theta \leqslant \alpha\), where \(\theta = \alpha\) is the equation of the tangent to the curve at \(O\). Find \(\alpha\) in terms of \(\pi\).
2. (a) If \(\mathrm { f } ( \theta ) = 1 - \sin 2 \theta\), show that \(\mathrm { f } \left( \frac { 1 } { 2 } ( 2 k + 1 ) \pi - \theta \right) = \mathrm { f } ( \theta )\) for all \(\theta\), where \(k\) is an integer.
   (b) Hence state the equations of the lines of symmetry of the curve $$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$
3. Sketch the curve with equation $$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$ State the maximum value of \(r\) and the corresponding values of \(\theta\).

4 A curve \(C\) has the cartesian equation \(x ^ { 3 } + y ^ { 3 } = a x y\), where \(x \geqslant 0 , y \geqslant 0\) and \(a > 0\).

1. Express the polar equation of \(C\) in the form \(r = \mathrm { f } ( \theta )\) and state the limits between which \(\theta\) lies. The line \(\theta = \alpha\) is a line of symmetry of \(C\).
2. Find and simplify an expression for \(\mathrm { f } \left( \frac { 1 } { 2 } \pi - \theta \right)\) and hence explain why \(\alpha = \frac { 1 } { 4 } \pi\).
3. Find the value of \(r\) when \(\theta = \frac { 1 } { 4 } \pi\).
4. Sketch the curve \(C\).

1

1. A curve has polar equation \(r = a \cos 3 \theta\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), where \(a\) is a positive constant.
   1. Sketch the curve, using a continuous line for sections where \(r > 0\) and a broken line for sections where \(r < 0\).
   2. Find the area enclosed by one of the loops.
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 3 } { 4 } } \frac { 1 } { \sqrt { 3 - 4 x ^ { 2 } } } \mathrm {~d} x\).
3. Use a trigonometric substitution to find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + 3 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\).

5 The limaçon of Pascal has polar equation \(r = 1 + 2 a \cos \theta\), where \(a\) is a constant.

1. Use your calculator to sketch the curve when \(a = 1\). (You need not distinguish between parts of the curve where \(r\) is positive and negative.)
2. By using your calculator to investigate the shape of the curve for different values of \(a\), positive and negative,
   (A) state the set of values of \(a\) for which the curve has a loop within a loop,
   (B) state, with a reason, the shape of the curve when \(a = 0\),
   (C) state what happens to the shape of the curve as \(a \rightarrow \pm \infty\),
   (D) name the feature of the curve that is evident when \(a = 0.5\), and find another value of \(a\) for which the curve has this feature.
3. Given that \(a > 0\) and that \(a\) is such that the curve has a loop within a loop, write down an equation for the values of \(\theta\) at which \(r = 0\). Hence show that the angle at which the curve crosses itself is \(2 \arccos \left( \frac { 1 } { 2 a } \right)\). Obtain the cartesian equations of the tangents at the point where the curve crosses itself. Explain briefly how these equations relate to the answer to part (ii)(A).

1

1. Given that \(y = \arctan \sqrt { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in terms of \(x\). Hence show that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x } ( x + 1 ) } \mathrm { d } x = \frac { \pi } { 2 }$$
2. A curve has cartesian equation $$x ^ { 2 } + y ^ { 2 } = x y + 1$$
   1. Show that the polar equation of the curve is $$r ^ { 2 } = \frac { 2 } { 2 - \sin 2 \theta }$$
   2. Determine the greatest and least positive values of \(r\) and the values of \(\theta\) between 0 and \(2 \pi\) for which they occur.
   3. Sketch the curve.

5 The points \(\mathrm { A } ( - 1,0 ) , \mathrm { B } ( 1,0 )\) and \(\mathrm { P } ( x , y )\) are such that the product of the distances PA and PB is 1 . You are given that the cartesian equation of the locus of P is $$\left( ( x + 1 ) ^ { 2 } + y ^ { 2 } \right) \left( ( x - 1 ) ^ { 2 } + y ^ { 2 } \right) = 1 .$$

1. Show that this equation may be written in polar form as $$r ^ { 4 } + 2 r ^ { 2 } = 4 r ^ { 2 } \cos ^ { 2 } \theta$$ Show that the polar equation simplifies to $$r ^ { 2 } = 2 \cos 2 \theta$$
2. Give a sketch of the curve, stating the values of \(\theta\) for which the curve is defined.
3. The equation in part (i) is now to be generalised to $$r ^ { 2 } = 2 \cos 2 \theta + k$$ where \(k\) is a constant.
   (A) Give sketches of the curve in the cases \(k = 1 , k = 2\). Describe how these two curves differ at the pole.
   (B) Give a sketch of the curve in the case \(k = 4\). What happens to the shape of the curve as \(k\) tends to infinity?
4. Sketch the curve for the case \(k = - 1\). What happens to the curve as \(k \rightarrow - 2\) ? \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

5 This question concerns the curves with polar equation $$r = \sec \theta + a \cos \theta ,$$ where \(a\) is a constant which may take any real value, and \(0 \leqslant \theta \leqslant 2 \pi\).

1. On a single diagram, sketch the curves for \(a = 0 , a = 1 , a = 2\).
2. On a single diagram, sketch the curves for \(a = 0 , a = - 1 , a = - 2\).
3. Identify a feature that the curves for \(a = 1 , a = 2 , a = - 1 , a = - 2\) share.
4. Name a distinctive feature of the curve for \(a = - 1\), and a different distinctive feature of the curve for \(a = - 2\).
5. Show that, in cartesian coordinates, equation (*) may be written $$y ^ { 2 } = \frac { a x ^ { 2 } } { x - 1 } - x ^ { 2 }$$ Hence comment further on the feature you identified in part (iii).
6. Show algebraically that, when \(a > 0\), the curve exists for \(1 < x < 1 + a\). Find the set of values of \(x\) for which the curve exists when \(a < 0\).

5 This question concerns curves with polar equation \(r = \sec \theta + a\), where \(a\) is a constant.

1. State the set of values of \(\theta\) between 0 and \(2 \pi\) for which \(r\) is undefined. For the rest of the question you should assume that \(\theta\) takes all values between 0 and \(2 \pi\) for which \(r\) is defined.
2. Use your graphical calculator to obtain a sketch of the curve in the case \(a = 0\). Confirm the shape of the curve by writing the equation in cartesian form.
3. Sketch the curve in the case \(a = 1\). Now consider the curve in the case \(a = - 1\). What do you notice?
   By considering both curves for \(0 < \theta < \pi\) and \(\pi < \theta < 2 \pi\) separately, describe the relationship between the cases \(a = 1\) and \(a = - 1\).
4. What feature does the curve exhibit for values of \(a\) greater than 1 ? Sketch a typical case.
5. Show that a cartesian equation of the curve \(r = \sec \theta + a\) is \(\left( x ^ { 2 } + y ^ { 2 } \right) ( x - 1 ) ^ { 2 } = a ^ { 2 } x ^ { 2 }\).

3 The function f is defined by \(\mathrm { f } ( x ) = \frac { 5 a x } { x ^ { 2 } + a ^ { 2 } }\), for \(x \in \mathbb { R }\) and \(a > 0\).

1. For the curve with equation \(y = \mathrm { f } ( x )\),
   1. write down the equation of the asymptote,
   2. find the range of values that \(y\) can take.
   3. For the curve with equation \(y ^ { 2 } = \mathrm { f } ( x )\), write down
      (a) the equation of the line of symmetry,
      (b) the maximum and minimum values of \(y\),
   4. the set of values of \(x\) for which the curve is defined.

7 A curve has polar equation \(r = 5 \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Sketch the curve, indicating the line of symmetry and stating the polar coordinates of the point \(P\) on the curve which is furthest away from the pole.
2. Calculate the area enclosed by the curve.
3. Find the cartesian equation of the tangent to the curve at \(P\).
4. Show that a cartesian equation of the curve is \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = ( 10 x y ) ^ { 2 }\).

2 A curve has polar equation \(r = \cos \theta \sin 2 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Find

1. the equations of the tangents at the pole,
2. the maximum value of \(r\),
3. a cartesian equation of the curve, in a form not involving fractions.

8 The equation of a curve is \(x ^ { 2 } + y ^ { 2 } - x = \sqrt { x ^ { 2 } + y ^ { 2 } }\).

1. Find the polar equation of this curve in the form \(r = \mathrm { f } ( \theta )\).
2. Sketch the curve.
3. The line \(x + 2 y = 2\) divides the region enclosed by the curve into two parts. Find the ratio of the two areas.

1 Find the cartesian equation corresponding to the polar equation \(r = ( \sqrt { } 2 ) \sec \left( \theta - \frac { 1 } { 4 } \pi \right)\). Sketch the the graph of \(r = ( \sqrt { } 2 ) \sec \left( \theta - \frac { 1 } { 4 } \pi \right)\), for \(- \frac { 1 } { 4 } \pi < \theta < \frac { 3 } { 4 } \pi\), indicating clearly the polar coordinates of the intersection with the initial line.

7 The diagram below shows the curve with polar equation \(r = a ( 1 - 2 \sin \theta )\) for \(0 \leqslant \theta \leqslant 2 \pi\), where \(a\) is a positive constant. \includegraphics[max width=\textwidth, alt={}, center]{76631941-3cd5-4b3e-a7e4-27b8f991975a-4_634_865_486_239} The curve crosses the initial line at A , and the points B and C are the lowest points on the two loops.

1. Find the values of \(r\) and \(\theta\) at the points A , B and C .
2. Find the set of values of \(\theta\) for the points on the inner loop (shown in the diagram with a broken line).

17 The curve \(C _ { 1 }\) has polar equation \(r = 2 a ( 1 + \sin \theta )\) for \(- \pi < \theta \leq \pi\) where \(a\) is a positive constant. \includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-22_469_830_402_605} The point \(M\) lies on \(C _ { 1 }\) and the initial line.
17

1. Write down, in terms of \(a\), the polar coordinates of \(M\) 17
2. \(\quad N\) is the point on \(C _ { 1 }\) that is furthest from the pole \(O\) Find, in terms of \(a\), the polar coordinates of \(N\) 17
3. The curve \(C _ { 2 }\) has polar equation \(r = 3 a\) for \(- \pi < \theta \leq \pi\) \(C _ { 2 }\) intersects \(C _ { 1 }\) at points \(P\) and \(Q\) Show that the area of triangle \(N P Q\) can be written in the form $$m \sqrt { 3 } a ^ { 2 }$$ where \(m\) is a rational number to be determined.
   17
4. On the initial line below, sketch the graph of \(r = 2 a ( 1 + \cos \theta )\) for \(- \pi < \theta \leq \pi\) Include the polar coordinates, in terms of \(a\), of any intersection points with the initial line.
   [0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-24_65_657_1425_991} \includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-25_2492_1721_217_150}

8 The curve \(C\) has the polar equation $$r = 4 - 2 \cos \theta \quad - \pi < \theta \leq \pi$$ 8

1. Verify that the point with polar coordinates \(\left( 3 , \frac { \pi } { 3 } \right)\) lies on \(C\) 8
2. Find the exact polar coordinates of the point on \(C\) which is furthest from the pole, \(O\) [3 marks]
   8
3. Find the exact Cartesian coordinates of the point on \(C\) where \(\theta\) is \(\frac { \pi } { 6 }\)

The curve \(C\) has polar equation \(r = a \cos 3\theta\), for \(-\frac{1}{6}\pi \leqslant \theta \leqslant \frac{1}{6}\pi\), where \(a\) is a positive constant.

1. Sketch \(C\). [2]
2. Find the area of the region enclosed by \(C\), showing full working. [3]
3. Using the identity \(\cos 3\theta \equiv 4\cos^3 \theta - 3\cos \theta\), find a cartesian equation of \(C\). [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}

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]

Sketch the polar graph of $$r = \sinh \theta + \cosh \theta$$ for \(0 \leq \theta \leq 2\pi\) \includegraphics{figure_11} [3 marks]

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]