jmc

From the given equation, $\frac{x^2+y^2}{xy}=6,$ so $x^2+y^2=6xy.$

Let $$a=\frac{x+y}{x-y}.$$Then $$a^2=\frac{x^2+2xy+y^2}{x^2-2xy+y^2}=\frac{8xy}{4xy}=2.$$Since $y>x>0,$ $a=\frac{x+y}{x-y}$ is negative. Therefore, $a=\boxed{-\sqrt{2}}.$