Step-by-Step Solution
Key Concept: The radical center equidistant from multiple circles has tangent distances that can be computed using angle addition formulas.
Since $AT$ and $BT$ are radical axes to $C_1$ and $C_2$ respectively, $T$ is the radical center. With $TA = TB = TD = 4$, we have $\tan\frac{\theta_1}{2} = \frac{2}{4} = \frac{1}{2}$ and $\tan\frac{\theta_2}{2} = \frac{3}{4}$. The radius of the circle through $C_1$ and $C_2$ is $r_3 = TA\tan\left(\frac{\theta_1 + \theta_2}{2}\right) = 4 \cdot \frac{\frac{2}{4} + \frac{3}{4}}{1 - \frac{2}{4} \cdot \frac{3}{4}} = 8$.
Correct Answer: 4