5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7f5fc83d-ab7c-4edb-a2c6-7a58f1357d5a-12_678_987_248_539}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of the curve \(C\) with equation
$$y ^ { 3 } - x ^ { 2 } + 4 x ^ { 2 } y = k$$
where \(k\) is a positive constant greater than 1
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
The point \(P\) lies on \(C\).
Given that the normal to \(C\) at \(P\) has equation \(y = x\), as shown in Figure 2, - find the value of \(k\).