# 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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