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\includegraphics[max width=\textwidth, alt={}, center]{9bbcee46-c5b8-4836-a4b4-f317bf8b1c0a-3_533_698_735_717}
The diagram shows the curve with parametric equations
$$x = 4 \sin \theta , \quad y = 1 + 3 \cos \left( \theta + \frac { 1 } { 6 } \pi \right)$$
for \(0 \leqslant \theta < 2 \pi\).
- Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed in the form \(k ( 1 + ( \sqrt { } 3 ) \tan \theta )\) where the exact value of \(k\) is to be determined.
- Find the equation of the normal to the curve at the point where the curve crosses the positive \(y\)-axis. Give your answer in the form \(y = m x + c\), where the constants \(m\) and \(c\) are exact.