There is a hyperbola $x^{2} - \frac{y^{2}}{35} = 1$ with foci at $\mathrm{F}(c, 0)$, $\mathrm{F}'(-c, 0)$ ($c > 0$). For a point P on this hyperbola in the first quadrant, let Q be a point on line $\mathrm{PF}'$ such that $\overline{\mathrm{PQ}} = \overline{\mathrm{PF}}$. When triangle $\mathrm{QF'F}$ and triangle $\mathrm{FF'P}$ are similar, the area of triangle PFQ is $\frac{q}{p}\sqrt{5}$. Find the value of $p + q$. (Here, $\overline{\mathrm{PF}'} < \overline{\mathrm{QF}'}$ and $p$ and $q$ are coprime natural numbers.) [4 points]
There is a hyperbola $x^{2} - \frac{y^{2}}{35} = 1$ with foci at $\mathrm{F}(c, 0)$, $\mathrm{F}'(-c, 0)$ ($c > 0$). For a point P on this hyperbola in the first quadrant, let Q be a point on line $\mathrm{PF}'$ such that $\overline{\mathrm{PQ}} = \overline{\mathrm{PF}}$.\\
When triangle $\mathrm{QF'F}$ and triangle $\mathrm{FF'P}$ are similar, the area of triangle PFQ is $\frac{q}{p}\sqrt{5}$. Find the value of $p + q$. (Here, $\overline{\mathrm{PF}'} < \overline{\mathrm{QF}'}$ and $p$ and $q$ are coprime natural numbers.) [4 points]