8 The diagram shows the sketch of part of a curve, the pole \(O\) and the initial line.
\includegraphics[max width=\textwidth, alt={}, center]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-20_609_670_358_703} The polar equation of the curve is \(r = 1 + \tan \theta\).
The point \(A\) is the point on the curve at which \(\theta = \frac { \pi } { 3 }\).
The perpendicular, \(A N\), from \(A\) to the initial line intersects the curve at the point \(B\).

1. Find the exact length of \(O A\).
2. 1. Given that, at the point \(B , \theta = \alpha\), show that \(( \cos \alpha + \sin \alpha ) ^ { 2 } = 1 + \frac { \sqrt { 3 } } { 2 }\).
   2. Hence, or otherwise, find \(\alpha\) in terms of \(\pi\).
3. Show that the area of triangle \(O A B\) is \(\frac { 3 + 2 \sqrt { 3 } } { 6 }\).
4. Find, in an exact simplified form, the area of the shaded region bounded by the curve and the line segment \(A B\).
   [0pt] [7 marks]
   \includegraphics[max width=\textwidth, alt={}]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-23_2488_1709_219_153}
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