Answer: 169

Solution:

Completing the square gives $\omega_1\colon (x+5)^2 + (y-12)^2 = 256$ and $\omega_2\colon (x-5)^2 + (y-12)^2 = 16,$ with centers $F_1 = (-5, 12)$ and $F_2 = (5, 12)$ and radii $16$ and $4.$ If a circle with center $P$ and radius $r$ is internally tangent to $\omega_1$ and externally tangent to $\omega_2,$ then $PF_1 = 16 - r$ and $PF_2 = 4 + r,$ so $PF_1 + PF_2 = 20.$

Thus $P$ lies on the ellipse with foci $F_1$ and $F_2$ and major axis $20:$ the semimajor axis is $10,$ the center-to-focus distance is $5,$ so the semiminor axis squared is $100 - 25 = 75,$ giving $$\frac{x^2}{100} + \frac{(y - 12)^2}{75} = 1,$$ i.e. $3x^2 + 4y^2 - 96y + 576 = 300.$ Substituting $y = ax$ yields $(3 + 4a^2)x^2 - 96ax + 276 = 0.$

The line $y = ax$ contains such a center exactly when this quadratic has a real root, i.e. $(96a)^2 - 4 \cdot 276\,(3 + 4a^2) \ge 0,$ which simplifies to $4800a^2 \ge 3312,$ so $a^2 \ge \frac{69}{100}.$ The smallest positive such $a$ has $m^2 = \frac{69}{100},$ and $p + q = 69 + 100 = 169.$