A circle with centre (2, 3) and radius 4 intersects the line $$x+y=3$$ at the points P and Q. If the tangents at P and Q intersect at the point $$S(\alpha,\beta)$$, then $$4\alpha-7\beta$$ is equal to ___________.

A triangle is formed by the tangents at the point (2, 2) on the curves $$y^2=2x$$ and $$x^2+y^2=4x$$, and the line $$x+y+2=0$$. If $$r$$ is the radius of its circumcircle, then $$r^2$$ is equal to ___________.

Let $$a_1=b_1=1$$ and $${a_n} = {a_{n - 1}} + (n - 1),{b_n} = {b_{n - 1}} + {a_{n - 1}},\forall n \ge 2$$. If $$S = \sum\limits_{n = 1}^{10} {{{{b_n}} \over {{2^n}}}} $$ and $$T = \sum\limits_{n = 1}^8 {{n \over {{2^{n - 1}}}}} $$, then $${2^7}(2S - T)$$ is equal to ____________.

The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is __________.