Consider a circle $$(x-\alpha)^2+(y-\beta)^2=50$$, where $$\alpha, \beta>0$$. If the circle touches the line $$y+x=0$$ at the point $$P$$, whose distance from the origin is $$4 \sqrt{2}$$, then $$(\alpha+\beta)^2$$ is equal to __________.

Two circles in the first quadrant of radii $$r_{1}$$ and $$r_{2}$$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $$x+y=2$$. Then $$r_{1}^{2}+r_{2}^{2}-r_{1} r_{2}$$ is equal to ___________.

Consider a circle $$C_{1}: x^{2}+y^{2}-4 x-2 y=\alpha-5$$. Let its mirror image in the line $$y=2 x+1$$ be another circle $$C_{2}: 5 x^{2}+5 y^{2}-10 f x-10 g y+36=0$$. Let $$r$$ be the radius of $$C_{2}$$. Then $$\alpha+r$$ is equal to _________.

Let the point $$(p, p+1)$$ lie inside the region $$E=\left\{(x, y): 3-x \leq y \leq \sqrt{9-x^{2}}, 0 \leq x \leq 3\right\}$$. If the set of all values of $$\mathrm{p}$$ is the interval $$(a, b)$$, then $$b^{2}+b-a^{2}$$ is equal to ___________.