Let the centre of a circle, passing through the points $$(0,0),(1,0)$$ and touching the circle $$x^2+y^2=9$$, be $$(h, k)$$. Then for all possible values of the coordinates of the centre $$(h, k), 4\left(h^2+k^2\right)$$ is equal to __________.

Consider two circles $$C_1: x^2+y^2=25$$ and $$C_2:(x-\alpha)^2+y^2=16$$, where $$\alpha \in(5,9)$$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $$C_1$$ and $$C_2$$ be $$\sin ^{-1}\left(\frac{\sqrt{63}}{8}\right)$$. If the length of common chord of $$C_1$$ and $$C_2$$ is $$\beta$$, then the value of $$(\alpha \beta)^2$$ equals _______.

Equations of two diameters of a circle are $$2 x-3 y=5$$ and $$3 x-4 y=7$$. The line joining the points $$\left(-\frac{22}{7},-4\right)$$ and $$\left(-\frac{1}{7}, 3\right)$$ intersects the circle at only one point $$P(\alpha, \beta)$$. Then, $$17 \beta-\alpha$$ is equal to _________.

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 __________.