Problem

The equation of a curve is $y = (5 - 2x)^{\frac{3}{2}} + 5$ for $x < \frac{5}{2}$.

(a) A point $P$ is moving along the curve in such a way that the $y$-coordinate of point $P$ is decreasing at $5$ units per second.

Find the rate at which the $x$-coordinate of point $P$ is increasing when $y = 32$.

(b) Point $A$ on the curve has $y$-coordinate $32$. Point $B$ on the curve is such that the gradient of the curve at $B$ is $-3$.

Find the equation of the perpendicular bisector of $AB$. Give your answer in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are integers.