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

7 The equation of a curve, in polar coordinates, is $$r = \sqrt { 3 } + \tan \theta , \quad \text { for } - \frac { 1 } { 3 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi$$

1. Find the equation of the tangent at the pole.
2. State the greatest value of \(r\) and the corresponding value of \(\theta\).
3. Sketch the curve.
4. Find the exact area of the region enclosed by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 4 } \pi\).

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 \(C\).

6 The equation of a curve in polar coordinates is \(r = \sin 5 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 5 } \pi\).

1. Sketch the curve and write down the equations of the tangents at the pole.
2. The line of symmetry meets the curve at the pole and at one other point \(A\). Find the equation of the line of symmetry and the cartesian coordinates of \(A\).
3. Find the area of the region enclosed by this curve.

4 The equation of a curve, in polar coordinates, is $$r = 2 \cos 2 \theta \quad ( - \pi < \theta \leqslant \pi ) .$$

1. Find the values of \(\theta\) which give the directions of the tangents at the pole. One loop of the curve is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{e4e1c424-8dd5-4d18-9950-e902de0301b0-3_362_720_653_708}
2. Find the exact value of the area of the region enclosed by the loop.

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 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.

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.

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 \(a < 0\).

1

1. 1. Differentiate the equation \(\sin y = x\) with respect to \(x\), and hence show that the derivative of \(\arcsin x\) is \(\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\).
   2. Evaluate the following integrals, giving your answers in exact form.
      (A) \(\int _ { - 1 } ^ { 1 } \frac { 1 } { \sqrt { 2 - x ^ { 2 } } } \mathrm {~d} x\) (B) \(\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { \sqrt { 1 - 2 x ^ { 2 } } } \mathrm {~d} x\)
2. A curve has polar equation \(r = \tan \theta , 0 \leqslant \theta < \frac { 1 } { 2 } \pi\). The points on the curve have cartesian coordinates \(( x , y )\). A sketch of the curve is given in Fig. 1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{99f0c663-bb5b-4456-854c-df177f5d8349-2_493_796_1123_605} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Show that \(x = \sin \theta\) and that \(r ^ { 2 } = \frac { x ^ { 2 } } { 1 - x ^ { 2 } }\).
   Hence show that the cartesian equation of the curve is $$y = \frac { x ^ { 2 } } { \sqrt { 1 - x ^ { 2 } } } .$$ Give the cartesian equation of the asymptote of the curve.

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 \(\left( x ^ { 2 } + y ^ { 2 } \right) ( x - 1 ) ^ { 2 } = a ^ { 2 } x ^ { 2 }\).

7 A curve has polar equation \(r = 1 + \cos 3 \theta\), for \(- \pi < \theta \leqslant \pi\).

1. Show that the line \(\theta = 0\) is a line of symmetry.
2. Find the equations of the tangents at the pole.
3. Find the exact value of the area of the region enclosed by the curve between \(\theta = - \frac { 1 } { 3 } \pi\) and \(\theta = \frac { 1 } { 3 } \pi\).

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 \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = ( 10 x y ) ^ { 2 }\).

9

1. It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } \theta \mathrm {~d} \theta$$ Show that, for \(n \geqslant 2\), $$n I _ { n } = ( n - 1 ) I _ { n - 2 } .$$
2. The equation of a curve, in polar coordinates, is $$r = \sin ^ { 3 } \theta , \quad \text { for } 0 \leqslant \theta \leqslant \pi$$
   1. Find the equations of the tangents at the pole and sketch the curve.
   2. Find the exact area of the region enclosed by the curve. RECOGNISING ACHIEVEMENT

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.

8 A curve has polar equation \(r = a ( 1 + \cos \theta )\), where \(a\) is a positive constant and \(0 \leqslant \theta < 2 \pi\).

1. Find the equation of the tangent at the pole.
2. Sketch the curve.
3. Find the area enclosed by the curve.

9 The equation of a curve in polar coordinates is \(r = 2 \sin 3 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).

1. Sketch the curve.
2. Find the area of the region enclosed by this curve.
3. By expressing \(\sin 3 \theta\) in terms of \(\sin \theta\), show that a cartesian equation for the curve is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } y - 2 y ^ { 3 } .$$ \section*{END OF QUESTION PAPER}

4 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$r = \theta + 2 \quad \text { and } \quad r = \theta ^ { 2 }$$ respectively, where \(0 \leqslant \theta \leqslant \pi\).

1. Find the polar coordinates of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
2. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
3. Find the area bounded by \(C _ { 1 } , C _ { 2 }\) and the line \(\theta = 0\).

5 Draw a sketch of the curve \(C\) whose polar equation is \(r = \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). On the same diagram draw the line \(\theta = \alpha\), where \(0 < \alpha < \frac { 1 } { 2 } \pi\). The region bounded by \(C\) and the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(R\). Find the exact value of \(\alpha\) for which the line \(\theta = \alpha\) divides \(R\) into two regions of equal area.

2 The curve \(C\) has polar equation $$r = a \left( 1 - \mathrm { e } ^ { - \theta } \right)$$ where \(a\) is a positive constant and \(0 \leqslant \theta < 2 \pi\).

1. Draw a sketch of \(C\).
2. Show that the area of the region bounded by \(C\) and the lines \(\theta = \ln 2\) and \(\theta = \ln 4\) is $$\frac { 1 } { 2 } a ^ { 2 } \left( \ln 2 - \frac { 13 } { 32 } \right)$$

11 The curve \(C\) has polar equation $$r = \frac { a } { 1 + \theta }$$ where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Show that \(r\) decreases as \(\theta\) increases.
2. The point \(P\) of \(C\) is further from the initial line than any other point of \(C\). Show that, at \(P\), $$\tan \theta = 1 + \theta$$ and verify that this equation has a root between 1.1 and 1.2.
3. Draw a sketch of \(C\).
4. Find the area of the region bounded by the initial line, the line \(\theta = \frac { 1 } { 2 } \pi\) and \(C\), leaving your answer in terms of \(\pi\) and \(a\).