The circles $x^2 + y^2 = 9$ and $x^2 + y^2 - 12y + 27 = 0$ touch each other. The equation of their common tangent is

$A$ 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 $........$.

If the point of contact of the circles $x^2+y^2-6x-4y+9=0$ and $x^2+y^2+2x+2y-7=0$ is $(\alpha, \beta)$, then $7\beta=$ (in $\alpha$)

If $P$ and $Q$ are the points of intersection of the circles $x^2 + y^2 + 3x + 7y + 2p - 5 = 0$ and $x^2 + y^2 + 2x + 2y - p^2 = 0$,then there is a circle passing through $P, Q$ and $(1, 1)$ for:

The length of the common chord of the circles $x^2+y^2-6x-4y+9=0$ and $x^2+y^2-8x-6y+23=0$ is

If tangents are drawn from the point $(5, -3)$ to the circle $x^2 + y^2 = 10$,then the equation of the chord of contact is: