16. The hyperbola \(C\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\).
- Show that an equation of the normal to \(C\) at the point \(P ( a \sec t , b \tan t )\) is
$$a x \sin t + b y = \left( a ^ { 2 } + b ^ { 2 } \right) \tan t .$$
The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(A\) and \(S\) is a focus of \(C\). Given that the eccentricity of \(C\) is \(\frac { 3 } { 2 }\), and that \(O A = 3 O S\), where \(O\) is the origin,
- determine the possible values of \(t\), for \(0 \leq t < 2 \pi\).
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[P5 June 2003 Qn 1]