$A(4,3), B(2,5)$ are two points. If $P$ is a variable point on the same side as that of the origin with respect to the line $A B$ and is at most at a distance of 5 units from the mid-point of $A B$, then the locus of $P$ is

The circles $x^2+y^2-2 x-4 y-4=0$ and $x^2+y^2+2 x+4 y-11=0$

If the line $4 x-3 y+7=0$ touches the circle $x^2+y^2-6 x+4 y-12=0$ at $(\alpha, \beta)$, then $\alpha+2 \beta=$

The slope of the common tangent drawn to the circles $x^2+y^2-4 x+12 y-216=0$ and $x^2+y^2+6 x-12 y+36=0$ is