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 Th cn \(C \mathbf { h }\) sp areq tin \(r = 2 + 2\) co \(\theta\), fo \(0 \leqslant \theta \leqslant \pi\).

1. Sk tch \(C\).
2. Fid \(b\) area 6 th reg œ \(n\) lo edy \(C\) ad \(b\) in tial lie .
3. Sh the the Cartesiare q tim \(C\) carb essed \(\mathrm { s } \left( 4 x ^ { 2 } + y ^ { 2 } \right) = \left( x ^ { 2 } + y ^ { 2 } - 2 x \right) ^ { 2 }\). \(\quad [ \beta\)

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 \(C\).

9 A curve has equation \(x ^ { 4 } + y ^ { 4 } = x ^ { 2 } + y ^ { 2 }\), where \(x\) and \(y\) are not both zero.

1. Show that the equation of the curve in polar coordinates is \(r ^ { 2 } = \frac { 2 } { 2 - \sin ^ { 2 } 2 \theta }\).
2. Deduce that no point on the curve \(x ^ { 4 } + y ^ { 4 } = x ^ { 2 } + y ^ { 2 }\) is further than \(\sqrt { 2 }\) from the origin.

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.

17 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]

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}