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If $$x > y > 0$$, $$x^2 + y^2 = 6xy$$ then $$\frac{x + y}{x - y}$$ is equal to
$$\left(\frac{x + y}{x - y}\right)^2 = \frac{x^2 + y^2 + 2xy}{x^2 + y^2 - 2xy} = \frac{8xy}{4xy} = 2$$. Since $$x > y > 0$$, the ratio is positive, so it equals $$\sqrt{2}$$.
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