Problem

A point $P$ is moving along the curve with equation $y = ax^{\frac{3}{2}} - 12x$ in such a way that the $x$-coordinate of $P$ is increasing at a constant rate of 5 units per second.

(a) Find the rate at which the $y$-coordinate of $P$ is changing when $x = 9$. Give your answer in terms of the constant $a$.

(b) Given that the curve has a minimum point when $x = \frac{1}{4}$, find the value of $a$.