When the axes are rotated through an angle $\theta$ about origin in anti-clockwise direction and then translated to the new origin $(2,-2)$, if the transformed equation the equation of $x^2+y^2=4$ is $X^2+Y^2+a X+b Y+c=0$ then $a+b+c=$

From a point $P(-4,0)$, two tangents are drawn to the circle $x^2+y^2-4 x-6 y-12=0$ touching the circle at $A$ and $B$. If the equation of the circle passing through $P, A$ and $B$ is $x^2+y^2+2 g x+2 f y+c=0$, then $(g, f)=$

If the equation of the polar of the point $(\alpha,-1)$ with respect to the circle $x^2+y^2-4 x-6 y-12=0$ is $y=\beta$, then $4(\alpha+\beta)=$

If $\theta$ is the angle between the tangents drawn from the point $(-1,-1)$ to the circle $x^2+y^2-4 x-6 y+c=0$ and $\cos \theta=-\frac{7}{25}$, then the radius of the circle is