A function $f$ is defined on the [complex numbers](https://artofproblemsolving.com/wiki/index.php/Complex_number) by $f(z)=(a+bi)z,$ where $a_{}$ and $b_{}$ are positive numbers.  This [function](https://artofproblemsolving.com/wiki/index.php/Function) has the property that the image of each point in the complex plane is [equidistant](https://artofproblemsolving.com/wiki/index.php/Equidistant) from that point and the [origin](https://artofproblemsolving.com/wiki/index.php/Origin).  Given that $|a+bi|=8$ and that $b^2=m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers,  find $m+n.$
Suppose we pick an arbitrary point on the [complex plane](https://artofproblemsolving.com/wiki/index.php/Complex_plane), say $(1,1)$. According to the definition of $f(z)$,  \[f(1+i) = (a+bi)(1+i) = (a-b) + (a+b)i,\] this image must be equidistant to $(1,1)$ and $(0,0)$. Thus the image must lie on the line with slope $-1$ and which passes through $\left(\frac 12, \frac12\right)$, so its graph is $x + y = 1$. Substituting $x = (a-b)$ and $y = (a+b)$, we get $2a = 1 \Rightarrow a = \frac 12$. 
By the [Pythagorean Theorem](https://artofproblemsolving.com/wiki/index.php/Pythagorean_Theorem), we have $\left(\frac{1}{2}\right)^2 + b^2 = 8^2 \Longrightarrow b^2 = \frac{255}{4}$, and the answer is $\boxed{259}$.