Polar curve intersection points

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)

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 \(\theta = 0\).

7 The diagram shows the sketch of a curve \(C _ { 1 }\).
\includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-18_362_734_360_635} The polar equation of the curve \(C _ { 1 }\) is $$r = 1 + \cos 2 \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$

1. Find the area of the region bounded by the curve \(C _ { 1 }\).
2. The curve \(C _ { 2 }\) whose polar equation is $$r = 1 + \sin \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$ intersects the curve \(C _ { 1 }\) at the pole \(O\) and at the point \(A\). The straight line drawn through \(A\) parallel to the initial line intersects \(C _ { 1 }\) again at the point \(B\).
   1. Find the polar coordinates of \(A\).
   2. Show that the length of \(O B\) is \(\frac { 1 } { 4 } ( \sqrt { 13 } + 1 )\).
   3. Find the length of \(A B\), giving your answer to three significant figures. \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-22_2486_1728_221_141}
      \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-23_2486_1728_221_141}
      \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-24_2488_1728_219_141}

2 Two polar curves, \(C _ { 1 }\) and \(C _ { 2 }\), are defined by \(C _ { 1 } : r = 2 \theta\) and \(C _ { 2 } : r = \theta + 1\) where \(0 \leqslant \theta \leqslant 2 \pi\). \(C _ { 1 }\) intersects the initial line at two points, the pole and the point \(A\).

1. Write down the polar coordinates of \(A\).
2. Determine the polar coordinates of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\). The diagram below shows a sketch of \(C _ { 1 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{007f07ee-cb29-4a97-93d9-2328079c4aea-2_681_1353_1318_244}
3. On the copy of this sketch in the Printed Answer Booklet, sketch \(C _ { 2 }\).

7 A curve \(C\) has polar equation $$r = 6 + 4 \cos \theta , \quad - \pi \leqslant \theta \leqslant \pi$$ The diagram shows a sketch of the curve \(C\), the pole \(O\) and the initial line.
\includegraphics[max width=\textwidth, alt={}, center]{0d894ac0-8d96-4182-8454-c306e1fdad8f-4_599_866_612_587}

1. Calculate the area of the region bounded by the curve \(C\).
2. The point \(P\) is the point on the curve \(C\) for which \(\theta = \frac { 2 \pi } { 3 }\). The point \(Q\) is the point on \(C\) for which \(\theta = \pi\).
   Show that \(Q P\) is parallel to the line \(\theta = \frac { \pi } { 2 }\).
3. The line \(P Q\) intersects the curve \(C\) again at a point \(R\). The line \(R O\) intersects \(C\) again at a point \(S\).
   1. Find, in surd form, the length of \(P S\).
   2. Show that the angle \(O P S\) is a right angle.

8 A curve has polar equation \(r = 3 + 2 \cos \theta\), where \(0 \leq \theta < 2 \pi\)
8

1. 1. State the maximum and minimum values of \(r\).
      [0pt] [2 marks]
      L
      8
2. (ii) Sketch the curve.
   \includegraphics[max width=\textwidth, alt={}, center]{e61d0202-49c9-4ed9-9fa3-f10734e17463-12_77_832_2037_651} 8
3. The curve \(r = 3 + 2 \cos \theta\) intersects the curve with polar equation \(r = 8 \cos ^ { 2 } \theta\), where \(0 \leq \theta < 2 \pi\) Find all of the points of intersection of the two curves in the form \([ r , \theta ]\).
   [0pt] [6 marks]