Question 9
A function $f$ is defined on the complex numbers by $f(z) = (a + bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a + bi| = 8$ and that $b^2 = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$