The curve \(C\) has polar equation \(r^2 = \frac{1}{\theta^2 + 1}\), for \(0 \leq \theta \leq \pi\).

1. Sketch \(C\) and state the polar coordinates of the point of \(C\) furthest from the pole.
2. Find the area of the region enclosed by \(C\), the initial line, and the half-line \(\theta = \pi\).
3. Show that, at the point of \(C\) furthest from the initial line, \(\left( \theta + \frac{1}{\theta} \right) \cot \theta - 1 = 0\) and verify that this equation has a root between 1.1 and 1.2.