A curve has cartesian equation \(x y = 8\). Show that the polar equation of the curve is \(r ^ { 2 } = 16 \operatorname { cosec } 2 \theta\).
The diagram shows a sketch of the curve, \(C\), whose polar equation is
$$r ^ { 2 } = 16 \operatorname { cosec } 2 \theta , \quad 0 < \theta < \frac { \pi } { 2 }$$
\includegraphics[max width=\textwidth, alt={}, center]{c4bce668-61f1-4be0-97ee-c635df7e1fc6-4_364_567_1635_726}
Find the polar coordinates of the point \(N\) which lies on the curve \(C\) and is closest to the pole \(O\).
The circle whose polar equation is \(r = 4 \sqrt { 2 }\) intersects the curve \(C\) at the points \(P\) and \(Q\). Find, in an exact form, the polar coordinates of \(P\) and \(Q\).
The obtuse angle \(P N Q\) is \(\alpha\) radians. Find the value of \(\alpha\), giving your answer to three significant figures.
(5 marks)