Please don't use just algebra

Algebra Level 5

Let \(a,b \in \mathbb{R}\) be fixed, with \(a<0\) and \(b>0\). Then suppose that \(f(\alpha, \beta)=\sqrt{(x-\alpha)^2+(y-\beta)^2}\) and some \(x,y \in \mathbb{R}\) give the minimum possible value of the expression below:

\(f(1,1)+f(-1,-1)+f(1,-1)+f(a,b)\)

Then we find that

\(x+y= 2\frac{a+b}{b-a+K}\)

Find K.

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