The curve C has polar equation \(r = a \sec^2 \theta\), where \(a\) is a positive constant and \(0 \leq \theta \leq \frac{1}{4} \pi\).

1. Sketch C, stating the polar coordinates of the point of intersection of C with the initial line and also with the half-line \(\theta = \frac{1}{4} \pi\).
2. Find the maximum distance of a point of C from the initial line.
3. Find the area of the region enclosed by C, the initial line and the half-line \(\theta = \frac{1}{4} \pi\).
4. Find, in the form \(y = f(x)\), the Cartesian equation of C.