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\includegraphics[max width=\textwidth, alt={}, center]{b9f29713-bc86-4869-9e54-195208e5e81d-4_511_609_264_769} The diagram shows the curve with equation, in polar coordinates, $$r = 3 + 2 \cos \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi .$$ The points \(P , Q , R\) and \(S\) on the curve are such that the straight lines \(P O R\) and \(Q O S\) are perpendicular, where \(O\) is the pole. The point \(P\) has polar coordinates ( \(r , \alpha\) ).

1. Show that \(O P + O Q + O R + O S = k\), where \(k\) is a constant to be found.
2. Given that \(\alpha = \frac { 1 } { 4 } \pi\), find the exact area bounded by the curve and the lines \(O P\) and \(O Q\) (shaded in the diagram).