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Q.
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
The given line is polar or $P(2, \beta)$ w.r.t. given circle
$x^2+y^2-4 x-6 y-3=0$
Chord or contact
$ \alpha x +\beta y -2( x +\alpha)-3( y +\beta)-3=0$
$ \Rightarrow(\alpha-2) x +(\beta-3) y -(2 \alpha+3 \beta+3)=0$....(i)
$\because$ But the equation of chord of contact is given
as : $x+y-3=0$...(ii)
comparing the coefficients
$\frac{\alpha-2}{1}=\frac{\beta-3}{1}=-\left(\frac{2 \alpha+3 \beta+3}{-3}\right)$
On solving $\alpha=-6$
$\beta=-5$
Now $ 4 \alpha-7 \beta=11$