Positive reals \(x, y\) satisfy \(x \neq y\) and \(\frac{x^2 + y^2}{xy} = k\). If replacing \(x\) by \(x + y\) and \(y\) by \(|x - y|\) leaves the value of \(k\) unchanged, then \(k\) equals

New value: ((x+y)^2 + (x-y)^2)/((x+y)|x-y|) = 2(x^2+y^2)/|x^2-y^2|. Equating to (x^2+y^2)/(xy) gives |x^2-y^2| = 2xy; with t = x/y > 1, t^2 - 2t - 1 = 0, t = 1+sqrt(2). k = t + 1/t = (1+sqrt2) + (sqrt2-1) = 2 sqrt(2).