2008 AIME II Problem 14

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.