17 The curve \(C _ { 1 }\) has polar equation \(r = 2 a ( 1 + \sin \theta )\) for \(- \pi < \theta \leq \pi\) where \(a\) is a positive constant.
\includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-22_469_830_402_605}
The point \(M\) lies on \(C _ { 1 }\) and the initial line.
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- Write down, in terms of \(a\), the polar coordinates of \(M\)
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- \(\quad N\) is the point on \(C _ { 1 }\) that is furthest from the pole \(O\)
Find, in terms of \(a\), the polar coordinates of \(N\)
17 - The curve \(C _ { 2 }\) has polar equation \(r = 3 a\) for \(- \pi < \theta \leq \pi\) \(C _ { 2 }\) intersects \(C _ { 1 }\) at points \(P\) and \(Q\)
Show that the area of triangle \(N P Q\) can be written in the form
$$m \sqrt { 3 } a ^ { 2 }$$
where \(m\) is a rational number to be determined.
17 - On the initial line below, sketch the graph of \(r = 2 a ( 1 + \cos \theta )\) for \(- \pi < \theta \leq \pi\) Include the polar coordinates, in terms of \(a\), of any intersection points with the initial line.
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[2 marks]
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\includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-25_2492_1721_217_150}