13.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-24_515_750_264_598}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
The curve \(C\) shown in Figure 4 has parametric equations
$$x = 1 + \sqrt { 3 } \tan \theta , \quad y = 5 \sec \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$
The curve \(C\) crosses the \(y\)-axis at \(A\) and has a minimum turning point at \(B\), as shown in Figure 4.
- Find the exact coordinates of \(A\).
- Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \lambda \sin \theta\), giving the exact value of the constant \(\lambda\).
- Find the coordinates of \(B\).
- Show that the cartesian equation for the curve \(C\) can be written in the form
$$y = k \sqrt { \left( x ^ { 2 } - 2 x + 4 \right) }$$
where \(k\) is a simplified surd to be found.