The curve C has polar equation \(r = a \tan\left(\frac{1}{8}\theta\right)\), where \(a\) is a positive constant and \(0 \leq \theta \leq 2\pi\).

(a) Sketch C and state, in terms of \(a\), the greatest distance of a point on C from the pole.

(b) Find, in terms of \(a\), the area of the region bounded by C and the initial line.

(c) Show that, at the point on C furthest from the initial line,

\(4 \sin\left(\frac{1}{4}\theta\right)\cos\theta + \sin\theta = 0\)

and verify that this equation has a root between 4.95 and 5.