8 The diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line. The curve \(C\) intersects the initial line at the point \(P\).
\includegraphics[max width=\textwidth, alt={}, center]{0eb3e96e-528c-4a99-b164-31cc865f0d68-20_432_949_402_525} The polar equation of \(C\) is \(r = \left( 1 - \tan ^ { 2 } \theta \right) \sec \theta , - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 }\).

1. Show that the area of the region bounded by the curve \(C\) is \(\frac { 8 } { 15 }\).
2. The curve whose polar equation is $$r = \frac { 1 } { 2 } \sec ^ { 3 } \theta , \quad - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 }$$ intersects \(C\) at the points \(A\) and \(B\).
   1. Find the polar coordinates of \(A\) and \(B\).
   2. Given that angle \(O A P =\) angle \(O B P = \alpha\), show that \(\tan \alpha = k \sqrt { 3 }\), where \(k\) is an integer.
   3. Using your value of \(k\) from part (b)(ii), state whether the point \(A\) lies inside or lies outside the circle whose diameter is \(O P\). Give a reason for your answer.
      [0pt] [1 mark]
      \includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-21_2484_1707_221_153}
      \includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-23_2484_1707_221_153}