System of equations..

From equation $(1):$

$\begin{aligned} \frac {x+y}{x-y} + \frac {x-y}{x+y} & = \frac 52 \\ \frac {(x+y)^2+(x-y)^2}{x^2-y^2} & = \frac 52 \\ \frac {2({\color{#3D99F6}x^2+y^2})}{x^2-y^2} & = \frac 52 & \small \color{#3D99F6} \text{Note }(2): \ x^2+y^2 = 90 \\ \frac {2({\color{#3D99F6}90})}{x^2-y^2} & = \frac 52 \\ \implies x^2-y^2 & = 72 & ...(1a) \end{aligned}$

Then $\begin{cases} (1a) + (2): & 2x^2 = 162\ & \implies x^2 = 81 & \implies x = \pm 9 \\ (1): & 81+y^2 = 90 & \implies y^2 = 9 & \implies y = \pm 3 \end{cases}$

Therefore, $\min (y-x) = -3-(+9) = \boxed{-12}$ .