Problem

Figure 4 shows a sketch of the curve $C$ with parametric equations

$$x=\sec t\qquad y=\sqrt{3}\tan\left(t+\frac{\pi}{3}\right)\qquad \frac{\pi}{6}<t<\frac{\pi}{2}$$

(a) Find $\dfrac{dy}{dx}$ in terms of $t$.

(b) Find an equation for the tangent to $C$ at the point where $t=\dfrac{\pi}{3}$.

Give your answer in the form $y=mx+c$, where $m$ and $c$ are constants.

(c) Show that all points on $C$ satisfy the equation

$$y=\frac{Ax^2+B\sqrt{3x^2-3}}{4-3x^2}$$

where $A$ and $B$ are constants to be found.