A circle with center $(2,3)$ 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 $\mathrm{S}(\mathrm{\alpha },\mathrm{\beta })$ ,  then $4\mathrm{\alpha }-7\mathrm{\beta }$ is equal to
___________.

The given line is the polar of $\mathrm{S}(\mathrm{\alpha },\mathrm{\beta })$ w.r.t given circle.
Circle :$x^2+y^2-4x-6y-3=0$
Chord of contact : $\mathrm{\alpha x}+\mathrm{\beta y}-2(\mathrm{x}+\mathrm{\alpha })-3(\mathrm{y}+\mathrm{\beta })-3=0$
$\Rightarrow (\mathrm{\alpha }-2)\mathrm{x}+(\mathrm{\beta }-3)\mathrm{y}-(2\mathrm{\alpha }+3\mathrm{\beta }+3)=0$
But equation of chord of contact is $\mathrm{x}+\mathrm{y}-3=0$
Comparing the coefficients
$\frac{\mathrm{\alpha }-2}{1}=\frac{\mathrm{\beta }-3}{1}=\frac{-(2\mathrm{\alpha }+3\mathrm{\beta }+3)}{-3}\Rightarrow \mathrm{\alpha }=-6,\mathrm{\beta }=-5\Rightarrow 4\mathrm{\alpha }-7\mathrm{\beta }=11$