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 ___________.

<p>The line $x + y = 3$ ..... (i)</p> <p>is polar of $S(\alpha ,\beta )$ w.r.t. circle</p> <p>${(x - 2)^2} + {(y - 3)^2} = 16$</p> <p>$\Rightarrow {x^2} + {y^2} - 4x - 6y - 3 = 0$</p> <p>Equation of polar is</p> <p>$\alpha x + \beta y - 2(x + \alpha ) - 3(4 + \beta ) - 3 = 0$</p> <p>$(\alpha - 2)x + (\alpha - 3)y - (2\alpha + 3\beta + 3) = 0$ ..... (ii)</p> <p>(i) and (ii) represent the same.</p> <p>$\therefore$ $${{\alpha - 2} \over 1} = {{\beta - 3} \over 1} = {{2\alpha + 3\beta + 3} \over 3}$$</p> <p>$\alpha - \beta + 1 = 0$</p> <p>$\alpha - 3\beta - 9 = 0$</p> <p>$\Rightarrow \alpha = - 6,\beta = - 5$</p> <p>$4\alpha - 7\beta = 11$</p>