Problem

The equation of a curve is

$$y = k\sqrt{4x + 1} - x + 5,$$

where $k$ is a positive constant.

(a) Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.

(b) Find the $x$-coordinate of the stationary point in terms of $k$.

(c) Given that $k = 10.5$, find the equation of the normal to the curve at the point where the tangent to the curve makes an angle of $\tan^{-1}(2)$ with the positive $x$-axis.