Let a and b be positive real numbers with a \geq b. Let \rho be the maximum possible value of \frac{a}{b} for which the system of equations
a^{2}+y^{2}=b^{2}+x^{2}=(a-x)^{2}+(b-y)^{2}
has a solution (x, y) satisfying 0 \leq x<a and 0 \leq y<b. Then \rho^{2} can be expressed as a fraction \frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.