(a) The polar equation is \(r = a \sec^2 \theta\). At \(\theta = 0\), \(r = a \sec^2(0) = a\), giving the point \((a, 0)\). At \(\theta = \frac{1}{4} \pi\), \(r = a \sec^2(\frac{1}{4} \pi) = 2a\), giving the point \((2a, \frac{1}{4} \pi)\).

(b) The maximum distance from the initial line is when \(\theta = \frac{1}{4} \pi\), where \(r = 2a\). The distance is \(y = r \sin \theta = 2a \sin(\frac{1}{4} \pi) = a\sqrt{2}\).

(c) The area \(A\) is given by \(\frac{1}{2} \int_0^{\frac{1}{4} \pi} r^2 \, d\theta\). Substituting \(r = a \sec^2 \theta\), we have:

\(A = \frac{1}{2} a^2 \int_0^{\frac{1}{4} \pi} \sec^4 \theta \, d\theta\)

\(= \frac{1}{2} a^2 \int_0^{\frac{1}{4} \pi} (\sec^2 \theta)(\tan^2 \theta + 1) \, d\theta\)

\(= \frac{1}{2} a^2 \left[ \tan \theta + \frac{1}{3} \tan^3 \theta \right]_0^{\frac{1}{4} \pi}\)

\(= \frac{3}{8} a^2\)

(d) The Cartesian form is derived by substituting \(x = r \cos \theta\) and \(y = r \sin \theta\). From \(r = a \sec^2 \theta\), we have:

\(x^2 + y^2 = r^2\)

\(r^2 \cos^2 \theta = a^2 \Rightarrow x^2 = a^2 - y^2\)

\(x^4 = a^2 (x^2 + y^2)\)

\(y = \sqrt{a^2 x^4 - x^2} = x \sqrt{a^2 x^2 - 1}\)