Polar curve with exponential function

Questions involving polar curves with exponential terms like r=ae^(-θ) or r=e^θ, often requiring integration or finding specific parameter values.

2 questions · Challenging +1.3

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CAIE Further Paper 1 2023 November Q6
13 marks Challenging +1.8
6 The curve \(C\) has polar equation \(r = \mathrm { e } ^ { - \theta } - \mathrm { e } ^ { - \frac { 1 } { 2 } \pi }\), where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  1. Sketch \(C\) and state, in exact form, the greatest distance of a point on \(C\) from the pole.
  2. Find the exact value of the area of the region bounded by \(C\) and the initial line.
  3. Show that, at the point on \(C\) furthest from the initial line, $$1 - e ^ { \theta - \frac { 1 } { 2 } \pi } - \tan \theta = 0$$ and verify that this equation has a root between 0.56 and 0.57 .
AQA FP3 2009 June Q7
14 marks Standard +0.8
7 The diagram shows the curve \(C _ { 1 }\) with polar equation $$r = 1 + 6 \mathrm { e } ^ { - \frac { \theta } { \pi } } , \quad 0 \leqslant \theta \leqslant 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{13cfb9ca-9495-4b69-80c5-9fb7e8e0f957-4_300_513_1414_760}
  1. Find, in terms of \(\pi\) and e , the area of the shaded region bounded by \(C _ { 1 }\) and the initial line.
  2. The polar equation of a curve \(C _ { 2 }\) is $$r = \mathrm { e } ^ { \frac { \theta } { \pi } } , \quad 0 \leqslant \theta \leqslant 2 \pi$$ Sketch the curve \(C _ { 2 }\) and state the polar coordinates of the end-points of this curve.
  3. The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the point \(P\). Find the polar coordinates of \(P\).