Question 14
Let $a$ and $b$ be positive real numbers with $a \ge b.$ Let $\rho$ be the maximum possible value of $\frac{a}{b}$ for which the system of equations $$
\begin{aligned} a^2 + y^2 &= b^2 + x^2 \\ &= (a - x)^2 + (b - y)^2 \end{aligned}
$$ has a solution $(x, y)$ satisfying $0 \le x \lt a$ and $0 \le y \lt b.$ Then $\rho^2$ can be expressed as a fraction $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
\begin{aligned} a^2 + y^2 &= b^2 + x^2 \\ &= (a - x)^2 + (b - y)^2 \end{aligned}
$$ has a solution $(x, y)$ satisfying $0 \le x \lt a$ and $0 \le y \lt b.$ Then $\rho^2$ can be expressed as a fraction $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$