16.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{824d73c5-525c-4876-ad66-33c8f1664277-44_742_673_248_696}
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\caption{Figure 6}
\end{figure}
Figure 6 shows a sketch of the curve \(C\) with parametric equations
$$x = 2 \tan t + 1 \quad y = 2 \sec ^ { 2 } t + 3 \quad - \frac { \pi } { 4 } \leqslant t \leqslant \frac { \pi } { 3 }$$
The line \(l\) is the normal to \(C\) at the point \(P\) where \(t = \frac { \pi } { 4 }\)
- Using parametric differentiation, show that an equation for \(l\) is
$$y = - \frac { 1 } { 2 } x + \frac { 17 } { 2 }$$
- Show that all points on \(C\) satisfy the equation
$$y = \frac { 1 } { 2 } ( x - 1 ) ^ { 2 } + 5$$
The straight line with equation
$$y = - \frac { 1 } { 2 } x + k \quad \text { where } k \text { is a constant }$$
intersects \(C\) at two distinct points.
- Find the range of possible values for \(k\).