4.09a Polar coordinates: convert to/from cartesian

5. (a) Sketch the curve with polar equation \(\quad r = 3 \cos 2 \theta , \quad - \frac { \pi } { 4 } \leq \theta < \frac { \pi } { 4 }\) (b) Find the area of the smaller finite region enclosed between the curve and the half-line $$\theta = \frac { \pi } { 6 }$$ (c) Find the exact distance between the two tangents which are parallel to the initial line.
(8)(Total 16 marks)

9. The diagram is a sketch of the two curves \(C _ { 1 }\) and \(C _ { 2 }\) with polar equations \(C _ { 1 } : r = 3 a ( 1 - \cos \theta ) , - \pi \leq \theta < \pi\) \(\mathrm { C } _ { 2 } : r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi\). \includegraphics[max width=\textwidth, alt={}, center]{8646b60a-3822-4d41-8978-1ccad1e216d6-2_318_776_1567_1082} 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 \(B\).
2. Show that the length of the line \(A B\) is \(\frac { 3 \sqrt { } 3 } { 2 } a\). The region inside \(C _ { 2 }\) and outside \(C _ { 1 }\) is shown shaded in the diagram above.
3. Find, in terms of \(a\), the area of this region. A badge is designed which has the shape of the shaded region.
   Given that the length of the line \(A B\) is 4.5 cm ,
4. calculate the area of this badge, giving your answer to three significant figures.
   (Total 16 marks)

4. The curve \(C\) has polar equation \(\quad r = 6 \cos \theta , \quad - \frac { \pi } { 2 } \leq \theta < \frac { \pi } { 2 }\), and the line \(D\) has polar equation \(\quad r = 3 \sec \left( \frac { \pi } { 3 } - \theta \right) , \quad - \frac { \pi } { 6 } < \theta < \frac { 5 \pi } { 6 }\).

1. Find a cartesian equation of \(C\) and a cartesian equation of \(D\).
2. Sketch on the same diagram the graphs of \(C\) and \(D\), indicating where each cuts the initial line. The graphs of \(C\) and \(D\) intersect at the points \(P\) and \(Q\).
3. Find the polar coordinates of \(P\) and \(Q\).
   (5)(Total 13 marks)

8. The curve \(C\) which passes through \(O\) has polar equation $$r = 4 a ( 1 + \cos \theta ) , \quad - \pi < \theta \leq \pi .$$ The line \(l\) has polar equation $$r = 3 a \sec \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 } .$$ The line \(l\) cuts \(C\) at the points \(P\) and \(Q\), as shown in the diagram.

1. Prove that \(P Q = 6 \sqrt { } 3 a\). The region \(R\), shown shaded in the diagram, is bounded by \(l\) and \(C\).
2. Use calculus to find the exact area of \(R\). \includegraphics[max width=\textwidth, alt={}, center]{d9aa1f75-ef35-4bf0-85c2-dff36872ca46-2_714_778_1959_1153}

4. \includegraphics[max width=\textwidth, alt={}, center]{d6befd60-de40-41b6-8ae5-48656dbca40c-3_535_1027_276_577} The diagram above shows a sketch of the curve \(C\) with polar equation $$r = 4 \sin \theta \cos ^ { 2 } \theta , \quad 0 \leq \theta < \frac { \pi } { 2 }$$ The tangent to \(C\) at the point \(P\) is perpendicular to the initial line.

1. Show that \(P\) has polar coordinates \(\left( \frac { 3 } { 2 } , \frac { \pi } { 6 } \right)\). The point \(Q\) on \(C\) has polar coordinates \(\left( \sqrt { 2 } , \frac { \pi } { 4 } \right)\).
   The shaded region \(R\) is bounded by \(O P , O Q\) and \(C\), as shown in the diagram above.
2. Show that the area of \(R\) is given by $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \left( \sin ^ { 2 } 2 \theta \cos 2 \theta + \frac { 1 } { 2 } - \frac { 1 } { 2 } \cos 4 \theta \right) \mathrm { d } \theta$$
3. Hence, or otherwise, find the area of \(R\), giving your answer in the form \(a + b \pi\), where \(a\) and \(b\) are rational numbers.
   (Total 14 marks)

8. (a) Sketch the curve \(C\) with polar equation $$r = 5 + \sqrt { 3 } \cos \theta , \quad 0 \leq \theta \leq 2 \pi$$ (b) Find the polar coordinates of the points where the tangents to \(C\) are parallel to the initial line \(\theta = 0\). Give your answers to 3 significant figures where appropriate.
(c) Using integration, find the area enclosed by the curve \(C\), giving your answer in terms of \(\pi\).

5. \begin{figure}[h] \end{figure} Figure 1 shows the curves given by the polar equations $$r = 2 , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 } ,$$ and $$r = 1.5 + \sin 3 \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$

1. Find the coordinates of the points where the curves intersect. The region \(S\), between the curves, for which \(r > 2\) and for which \(r < ( 1.5 + \sin 3 \theta )\), is shown shaded in Figure 1.
2. Find, by integration, the area of the shaded region \(S\), giving your answer in the form \(a \pi + b \sqrt { 3 }\), where \(a\) and \(b\) are simplified fractions.

2. The curve \(C\) has polar equation $$r = 1 + 2 \cos \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ At the point \(P\) on \(C\), the tangent to \(C\) is parallel to the initial line.
Given that \(O\) is the pole, find the exact length of the line \(O P\).

9. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curves given by the polar equations $$r = 1 \text { and } r = 2 - 2 \sin \theta$$

1. Find the coordinates of the points where the curves intersect. The region \(S\), between the curves, for which \(r < 1\) and for which \(r < 2 - 2 \sin \theta\), is shown shaded in Figure 1.
2. Find, by integration, the area of the shaded region \(S\), giving your answer in the form \(a \pi + b \sqrt { } 3\), where \(a\) and \(b\) are rational numbers.

6. The curve \(C\) has polar equation $$r ^ { 2 } = a ^ { 2 } \cos 2 \theta , \quad \frac { - \pi } { 4 } \leq \theta \leq \frac { \pi } { 4 }$$

1. Sketch the curve \(C\).
2. Find the polar coordinates of the points where tangents to \(C\) are parallel to the initial line.
3. Find the area of the region bounded by \(C\).

1

1. A curve has polar equation \(r = a ( \sqrt { 2 } + 2 \cos \theta )\) for \(- \frac { 3 } { 4 } \pi \leqslant \theta \leqslant \frac { 3 } { 4 } \pi\), where \(a\) is a positive constant.
   1. Sketch the curve.
   2. Find, in an exact form, the area of the region enclosed by the curve.
2. 1. Find the Maclaurin series for the function \(\mathrm { f } ( x ) = \tan \left( \frac { 1 } { 4 } \pi + x \right)\), up to the term in \(x ^ { 2 }\).
   2. Use the Maclaurin series to show that, when \(h\) is small, $$\int _ { - h } ^ { h } x ^ { 2 } \tan \left( \frac { 1 } { 4 } \pi + x \right) \mathrm { d } x \approx \frac { 2 } { 3 } h ^ { 3 } + \frac { 4 } { 5 } h ^ { 5 }$$

1

1. A curve has polar equation \(r = a ( 1 - \cos \theta )\), where \(a\) is a positive constant.
   1. Sketch the curve.
   2. Find the area of the region enclosed by the section of the curve for which \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\) and the line \(\theta = \frac { 1 } { 2 } \pi\).
2. Use a trigonometric substitution to show that \(\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { 1 } { 4 \sqrt { 3 } }\).
3. In this part of the question, \(\mathrm { f } ( x ) = \arccos ( 2 x )\).
   1. Find \(\mathrm { f } ^ { \prime } ( x )\).
   2. Use a standard series to expand \(\mathrm { f } ^ { \prime } ( x )\), and hence find the series for \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to the term in \(x ^ { 5 }\).

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 \(\sum _ { r = 0 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) 4 ^ { r } } = \ln 3\).

1

1. 1. Given that \(\mathrm { f } ( t ) = \arcsin t\), write down an expression for \(\mathrm { f } ^ { \prime } ( t )\) and show that $$\mathrm { f } ^ { \prime \prime } ( t ) = \frac { t } { \left( 1 - t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$$
   2. Show that the Maclaurin expansion of the function \(\arcsin \left( x + \frac { 1 } { 2 } \right)\) begins $$\frac { \pi } { 6 } + \frac { 2 } { \sqrt { 3 } } x$$ and find the term in \(x ^ { 2 }\).
2. Sketch the curve with polar equation \(r = \frac { \pi a } { \pi + \theta }\), where \(a > 0\), for \(0 \leqslant \theta < 2 \pi\). Find, in terms of \(a\), the area of the region bounded by the part of the curve for which \(0 \leqslant \theta \leqslant \pi\) and the lines \(\theta = 0\) and \(\theta = \pi\).
3. Find the exact value of the integral $$\int _ { 0 } ^ { \frac { 3 } { 2 } } \frac { 1 } { 9 + 4 x ^ { 2 } } \mathrm {~d} x$$

9 The equation of a curve, in polar coordinates, is $$r = \sec \theta + \tan \theta , \quad \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi$$

1. Sketch the curve.
2. Find the exact area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 3 } \pi\).
3. Find a cartesian equation of the curve.

4 The equation of a curve, in polar coordinates, is $$r = 1 + 2 \sec \theta , \quad \text { for } - \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi$$

1. Find the exact area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 6 } \pi\). [The result \(\int \sec \theta \mathrm { d } \theta = \ln | \sec \theta + \tan \theta |\) may be assumed.]
2. Show that a cartesian equation of the curve is \(( x - 2 ) \sqrt { x ^ { 2 } + y ^ { 2 } } = x\).

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 \(r = a \mathrm { e } ^ { - k \theta }\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) and \(k\) are positive constants. The points A and B on the curve correspond to \(\theta = 0\) and \(\theta = \pi\) respectively.
   1. Sketch the curve.
   2. Find the area of the region enclosed by the curve and the line AB .
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { 3 + 4 x ^ { 2 } } \mathrm {~d} x\).
3. 1. Find the Maclaurin series for \(\tan x\), up to the term in \(x ^ { 3 }\).
   2. Use this Maclaurin series to show that, when \(h\) is small, \(\int _ { h } ^ { 4 h } \frac { \tan x } { x } \mathrm {~d} x \approx 3 h + 7 h ^ { 3 }\).

5 Cartesian coordinates \(( x , y )\) and polar coordinates \(( r , \theta )\) are set up in the usual way, with the pole at the origin and the initial line along the positive \(x\)-axis, so that \(x = r \cos \theta\) and \(y = r \sin \theta\). A curve has polar equation \(r = k + \cos \theta\), where \(k\) is a constant with \(k \geqslant 1\).

1. Use your graphical calculator to obtain sketches of the curve in the three cases $$k = 1 , k = 1.5 \text { and } k = 4$$
2. Name the special feature which the curve has when \(k = 1\).
3. For each of the three cases, state the number of points on the curve at which the tangent is parallel to the \(y\)-axis.
4. Express \(x\) in terms of \(k\) and \(\theta\), and find \(\frac { \mathrm { d } x } { \mathrm {~d} \theta }\). Hence find the range of values of \(k\) for which there are just two points on the curve where the tangent is parallel to the \(y\)-axis. The distance between the point ( \(r , \theta\) ) on the curve and the point ( 1,0 ) on the \(x\)-axis is \(d\).
5. Use the cosine rule to express \(d ^ { 2 }\) in terms of \(k\) and \(\theta\), and deduce that \(k ^ { 2 } \leqslant d ^ { 2 } \leqslant k ^ { 2 } + 1\).
6. Hence show that, when \(k\) is large, the shape of the curve is very nearly circular.

7. \includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-5_648_1590_296_275} The circle \(C _ { 1 }\) has centre \(O\) and radius \(R\). The tangents \(A P\) and \(B P\) to \(C _ { 1 }\) meet at the point \(P\) and angle \(A P B = 2 \alpha , 0 < \alpha < \frac { \pi } { 2 }\). A sequence of circles \(C _ { 1 } , C _ { 2 } , \ldots , C _ { n } , \ldots\) is drawn so that each new circle \(C _ { n + 1 }\) touches each of \(C _ { n } , A P\) and \(B P\) for \(n = 1,2,3 , \ldots\) as shown in Figure 2. The centre of each circle lies on the line \(O P\).

1. Show that the radii of the circles form a geometric sequence with common ratio $$\frac { 1 - \sin \alpha } { 1 + \sin \alpha }$$
2. Find, in terms of \(R\) and \(\alpha\), the total area enclosed by all the circles, simplifying your answer. The area inside the quadrilateral \(P A O B\), not enclosed by part of \(C _ { 1 }\) or any of the other circles, is \(S\).
3. Show that $$S = R ^ { 2 } \left( \alpha + \cot \alpha - \frac { \pi } { 4 } \operatorname { cosec } \alpha - \frac { \pi } { 4 } \sin \alpha \right) .$$
4. Show that, as \(\alpha\) varies, $$\frac { \mathrm { d } S } { \mathrm {~d} \alpha } = R ^ { 2 } \cot ^ { 2 } \alpha \left( \frac { \pi } { 4 } \cos \alpha - 1 \right)$$
5. Find, in terms of \(R\), the least value of \(S\) for \(\frac { \pi } { 6 } \leq \alpha \leq \frac { \pi } { 4 }\).

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curves with polar equations $$\begin{array} { l l } r = 2 \sin \theta & 0 \leqslant \theta \leqslant \pi \\ r = 1.5 - \sin \theta & 0 \leqslant \theta \leqslant 2 \pi \end{array}$$ The curves intersect at the points \(P\) and \(Q\).

1. Find the polar coordinates of the point \(P\) and the polar coordinates of the point \(Q\). The region \(R\), shown shaded in Figure 1, is enclosed by the two curves.
2. Find the exact area of \(R\), giving your answer in the form \(p \pi + q \sqrt { 3 }\), where \(p\) and \(q\) are rational numbers to be found.

3

1. A curve has polar equation \(r = a \tan \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\), where \(a\) is a positive constant.
   1. Sketch the curve.
   2. Find the area of the region between the curve and the line \(\theta = \frac { 1 } { 4 } \pi\). Indicate this region on your sketch.
2. 1. Find the eigenvalues and corresponding eigenvectors for the matrix \(\mathbf { M }\) where $$\mathbf { M } = \left( \begin{array} { l l } 0.2 & 0.8 \\ 0.3 & 0.7 \end{array} \right)$$
   2. Give a matrix \(\mathbf { Q }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } = \mathbf { Q D } \mathbf { Q } ^ { - 1 }\). Section B (18 marks)

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.