16.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d37eaba2-0a25-4abf-b2c8-1e08673229fb-26_1241_1130_251_440}
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\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of the curve \(C\) with parametric equations \(x = - 3 + 6 \sin \theta , \quad y = 9 \cos 2 \theta \quad - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 4 }\) where \(\theta\) is a parameter.
a. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\)
The line \(l\) is normal to \(C\) at the point \(P\) where \(\theta = \frac { \pi } { 6 }\)
b. Show that an equation for \(l\) is
$$y = \frac { 1 } { 3 } x + \frac { 9 } { 2 }$$
c. The cartesian equation for the curve \(C\) can be written in the form
$$y = a - \frac { 1 } { 2 } ( x + b ) ^ { 2 }$$
where \(a\) and \(b\) are integers to be found.
The straight line with equation
$$y = \frac { 1 } { 3 } x + k$$
where \(k\) is a constant intersects \(C\) at two distinct points.
d. Find the range of possible values for \(k\).