2014 AIME I Problem 7

Let w and z be complex numbers such that |w|=1 and |z|=10. Let \theta= \arg \left(\frac{w-z}{z}\right). The maximum possible value of \tan ^{2} \theta can be written as \frac{p}{q}, where p and q are relatively prime positive integers. Find p+q. (Note that \arg (w), for w \neq 0, denotes the measure of the angle that the ray from 0 to w makes with the positive real axis in the complex plane.)