Show polar curve has Cartesian form

7 The curve \(C _ { 1 }\) has polar equation \(r = a ( \cos \theta + \sin \theta )\) for \(- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 3 } { 4 } \pi\), where \(a\) is a positive constant.

1. Find a Cartesian equation for \(C _ { 1 }\) and show that it represents a circle, stating its radius and the Cartesian coordinates of its centre.
2. Sketch \(C _ { 1 }\) and state the greatest distance of a point on \(C _ { 1 }\) from the pole. \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-14_2721_40_107_2010} The curve \(C _ { 2 }\) with polar equation \(r = a \theta\) intersects \(C _ { 1 }\) at the pole and the point with polar coordinates \(( a \phi , \phi )\).
3. Verify that \(1.25 < \phi < 1.26\).
4. Show that the area of the smaller region enclosed by \(C _ { 1 }\) and \(C _ { 2 }\) is equal to $$\frac { 1 } { 2 } a ^ { 2 } \left( \frac { 3 } { 4 } \pi + \frac { 1 } { 3 } \phi ^ { 3 } - \phi + \frac { 1 } { 2 } \cos 2 \phi \right)$$ and deduce, in terms of \(a\) and \(\phi\), the area of the larger region enclosed by \(C _ { 1 }\) and \(C _ { 2 }\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

3 The curve \(C\) has polar equation \(r = 2 + 2 \cos \theta\), for \(0 \leqslant \theta \leqslant \pi\).

1. Sketch \(C\).
2. Find the area of the region enclosed by \(C\) and the initial line.
3. Show that the Cartesian equation of \(C\) can be expressed as \(4 \left( x ^ { 2 } + y ^ { 2 } \right) = \left( x ^ { 2 } + y ^ { 2 } - 2 x \right) ^ { 2 }\).

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 \(9 y ^ { 2 } = \mathrm { f } ( x )\).

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 \(C\).

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.

16 The curve \(C\) has the polar equation $$r = \frac { 2 } { \sqrt { \cos ^ { 2 } \theta + 4 \sin ^ { 2 } \theta } } \quad - \pi < \theta \leq \pi$$ 16

1. Show that the Cartesian equation of \(C\) can be written as $$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(a\) and \(b\) are positive integers to be determined.
   [0pt] [4 marks]
   16
2. Hence sketch the graph of \(C\) on the axes below. Indicate the value of any intercepts of the curve with the axes. \includegraphics[max width=\textwidth, alt={}, center]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-23_1122_1121_452_447}

6 The curve \(P\) has polar equation \(r = \frac { 1 } { 1 - \sin \theta }\) for \(0 \leqslant \theta < 2 \pi , \theta \neq \frac { 1 } { 2 } \pi\).

1. Determine, in the form \(y = \mathrm { f } ( x )\), the cartesian equation of \(P\).
2. Sketch \(P\).
3. Evaluate \(\int _ { \pi } ^ { 2 \pi } \frac { 1 } { ( 1 - \sin \theta ) ^ { 2 } } \mathrm {~d} \theta\).

Fig. 5 shows a circle with centre C \((a, 0)\) and radius \(a\). B is the point \((0, 1)\). The line BC intersects the circle at P and Q. P is above the \(x\)-axis and Q is below. \includegraphics{figure_5}

1. Show that, in the case \(a = 1\), P has coordinates \(\left(1 - \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\). Write down the coordinates of Q. [3]
2. Show that, for all positive values of \(a\), the coordinates of P are $$x = a\left(1 - \frac{a}{\sqrt{a^2 + 1}}\right), \quad y = \frac{a}{\sqrt{a^2 + 1}} \quad (*)$$ Write down the coordinates of Q in a similar form. [4] Now let the variable point P be defined by the parametric equations (*) for all values of the parameter \(a\), positive, zero and negative. Let Q be defined for all \(a\) by your answer in part (ii).
3. Using your calculator, sketch the locus of P as \(a\) varies. State what happens to P as \(a \to \infty\) and as \(a \to -\infty\). Show algebraically that this locus has an asymptote at \(y = -1\). On the same axes, sketch, as a dotted line, the locus of Q as \(a\) varies. [8] (The single curve made up of these two loci and including the point B is called a right strophoid.)
4. State, with a reason, the size of the angle POQ in Fig. 5. What does this indicate about the angle at which a right strophoid crosses itself? [3]

The polar equation of the circle \(C\) is $$r = a(\cos \theta + \sin \theta)$$ Find, in terms of \(a\), the radius of \(C\). Fully justify your answer. [4 marks]

The curve C has polar equation $$r^2 \sin 2\theta = 4$$ Find a Cartesian equation for C. Circle your answer. [1 mark] \(y = 2x\) \quad \(y = \frac{x}{2}\) \quad \(y = \frac{2}{x}\) \quad \(y = 4x\)

The curve C has polar equation $$r = \frac{A}{5 + 3 \cos \theta} \quad (-\pi < \theta \leq \pi)$$

1. Show that \(r\) takes values in the range \(\frac{1}{k} \leq r \leq k\), where \(k\) is an integer. [2 marks]
2. Find the Cartesian equation of C in the form \(y^2 = f(x)\) [4 marks]
3. The ellipse E has equation $$y^2 + \frac{16x^2}{25} = 1$$ Find the transformation that maps the graph of E onto C [4 marks]