9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f577461-24b7-4615-b58b-e67597fd9675-28_597_1020_251_525}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Figure 4 shows a sketch of the curve \(C\) with parametric equations
$$x = \sec t \quad y = \sqrt { 3 } \tan \left( t + \frac { \pi } { 3 } \right) \quad \frac { \pi } { 6 } < t < \frac { \pi } { 2 }$$
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\)
- Find an equation for the tangent to \(C\) at the point where \(t = \frac { \pi } { 3 }\)
Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
- Show that all points on \(C\) satisfy the equation
$$y = \frac { A x ^ { 2 } + B \sqrt { 3 x ^ { 2 } - 3 } } { 4 - 3 x ^ { 2 } }$$
where \(A\) and \(B\) are constants to be found.