Problem

Figure 2 shows a sketch of the curve $C$ with equation

$$y^3-x^2+4x^2y=k$$

where $k$ is a positive constant greater than $1$

(a) Find

$$\frac{dy}{dx}$$

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,

(b) find the value of $k$.