Dynamic Geometry: P99

Let the center of the unit semicircle be $O(0,0)$ , the origin of the $xy$ -plane, and its diameter $AB$ , the center of the purple circle be $P$ , $\angle POB = \theta$ , and $t = \tan \frac \theta 2$ . Also let an arbitrary point on the locus or the tangent point of the purple and orange circles be $T_2(x,y)$ . From the calculations in Dynamic Geometry: P80 Series, we have

$\begin{cases} x = \dfrac {5-t^2}{t^2+2t+5} \\ y = \dfrac {4t}{t^2+2t+5} \end{cases}$

Then the area under the locus is given by:

$\begin{aligned} A & = \int_0^{t=\sqrt 5} x \ \ce dy \\ & = \int_0^{t=\sqrt 5} \frac {5-t^2}{t^2+2t+5} \ \ce d \left(\frac {4t}{t^2+2t+5} \right) \\ & = \int_0^{\sqrt 5} \frac {4(5-t^2)^2}{(t^2+2t+5)^3} dt \\ & = \frac 54 \tan^{-1} 2 - \frac 12 \end{aligned}$

Therefore $p+q+m+n = 5+4+2+1 = \boxed{12}$ .