The curve \(C\) has polar equation
\(r=\frac{a}{1+\theta},\)
where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac{1}{2} \pi\).
(i) Show that \(r\) decreases as \(\theta\) increases.

(ii) The point \(P\) of \(C\) is further from the initial line than any other point of \(C\). Show that, at \(P\),
\(\tan \theta=1+\theta,\)
and verify that this equation has a root between 1.1 and 1.2.

(iii) Draw a sketch of \(C\).

(iv) Find the area of the region bounded by the initial line, the line \(\theta=\frac{1}{2} \pi\) and \(C\), leaving your answer in terms of \(\pi\) and \(a\).