If the circles $${x^2} + {y^2} + 6x + 8y + 16 = 0$$ and $${x^2} + {y^2} + 2\left( {3 - \sqrt 3 } \right)x + 2\left( {4 - \sqrt 6 } \right)y = k + 6\sqrt 3 + 8\sqrt 6 $$, $$k > 0$$, touch internally at the point $$P(\alpha ,\beta )$$, then $${\left( {\alpha + \sqrt 3 } \right)^2} + {\left( {\beta + \sqrt 6 } \right)^2}$$ is equal to ________________.

If one of the diameters of the circle $${x^2} + {y^2} - 2\sqrt 2 x - 6\sqrt 2 y + 14 = 0$$ is a chord of the circle $${(x - 2\sqrt 2 )^2} + {(y - 2\sqrt 2 )^2} = {r^2}$$, then the value of r^{2} is equal to ____________.

Let the lines $$y + 2x = \sqrt {11} + 7\sqrt 7 $$ and $$2y + x = 2\sqrt {11} + 6\sqrt 7 $$ be normal to a circle $$C:{(x - h)^2} + {(y - k)^2} = {r^2}$$. If the line $$\sqrt {11} y - 3x = {{5\sqrt {77} } \over 3} + 11$$ is tangent to the circle C, then the value of $${(5h - 8k)^2} + 5{r^2}$$ is equal to __________.

Let a circle C of radius 5 lie below the x-axis. The line L_{1} : 4x + 3y + 2 = 0 passes through the centre P of the circle C and intersects the line L_{2} = 3x $$-$$ 4y $$-$$ 11 = 0 at Q. The line L_{2} touches C at the point Q. Then the distance of P from the line 5x $$-$$ 12y + 51 = 0 is ______________.