Problem Analysis #28

Consider two points $A(5,0)$ and $B(-5,0)$ in the coordinate plane. A point $C$ and a point $P$ in the same plane satisfy the following conditions:$\begin{aligned}\text{(i)} &\quad \overline{\text{AC}} = 6,\\\text{(ii)} &\quad \text{A point P lies on segment } AC,\\\text{(iii)} &\quad \overline{\text{BP}} + \overline{\text{CP}} = 8.\end{aligned}$Let $\angle A = \theta$ in triangle $ABC$, and let the area of triangle $BCP$ be denoted by $f(\theta)$.For an angle $\beta$ satisfying$\cos \beta = \frac{4}{5}, \qquad 0 < \beta < \frac{\pi}{2},$assume that$f'(\beta) = -\frac{p}{q},$where $p$ and $q$ are relatively prime positive integers.(Also, point $C$ does not lie on segment $AB$, and point $P$ is distinct from point $C$.)Find the value of$p + q^{2}.$oindent