Find constant from given area

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with polar equation $$r = a + 3 \cos \theta , \quad a > 0 , \quad 0 \leqslant \theta < 2 \pi$$ The area enclosed by the curve is \(\frac { 107 } { 2 } \pi\).
Find the value of \(a\).

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with polar equation $$r = a + 3 \cos \theta , \quad a > 0 , \quad 0 \leqslant \theta < 2 \pi$$ The area enclosed by the curve is \(\frac { 107 } { 2 } \pi\).
Find the value of \(a\).

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve with polar equation $$r = 6 + a \sin \theta$$ where \(0 < a < 6\) and \(0 \leqslant \theta < 2 \pi\)
The area enclosed by the curve is \(\frac { 97 \pi } { 2 }\)
Find the value of the constant \(a\).

12 For any positive parameter \(k\), the curve \(C _ { k }\) is defined by the polar equation
\(\mathrm { r } = \mathrm { k } ( \cos \theta + 1 ) + \frac { 10 } { \mathrm { k } } , 0 \leqslant \theta \leqslant 2 \pi\).
For each value of \(k\) the curve is a single, closed loop with no self-intersections. The diagram shows \(C _ { 10.5 }\) for the purpose of illustration.
\includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-6_558_723_550_242} Each curve, \(C _ { k }\), encloses a certain area, \(A _ { k }\).
You are given that there is a single minimum value of \(A _ { k }\).
Determine, in an exact form, the value of \(k\) for which \(C _ { k }\) encloses this minimum area.

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} \begin{figure}[h] \end{figure} Figure 1 shows the design for a bathing pool.
The pool, \(P\), shown unshaded in Figure 1, is surrounded by a tiled area, \(T\), shown shaded in Figure 1. The tiled area is bounded by the edge of the pool and by a circle, \(C\), with radius 6 m .
The centre of the pool and the centre of the circle are the same point.
The edge of the pool is modelled by the curve with polar equation $$r = 4 - a \sin 3 \theta \quad 0 \leqslant \theta \leqslant 2 \pi$$ where \(a\) is a positive constant.
Given that the shortest distance between the edge of the pool and the circle \(C\) is 0.5 m ,

1. determine the value of \(a\).
2. Hence, using algebraic integration, determine, according to the model, the exact area of \(T\).

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with polar equation $$r = a + 3 \cos \theta , \quad a > 0 , \quad 0 \leq \theta < 2 \pi .$$ The area enclosed by the curve is \(\frac { 107 } { 2 } \pi\).
Find the value of \(a\).