16.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{d37eaba2-0a25-4abf-b2c8-1e08673229fb-26_1241_1130_251_440}
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\caption{Figure 3}
\end{center}
\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 $k$.

\hfill \mbox{\textit{Edexcel PMT Mocks  Q16 [12]}}