Question 7
Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10.$ Let $\theta = \arg\left(\tfrac{w-z}{z}\right).$ 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 $w \ne 0,$ denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane.)