Suppose the tangents drawn to the circle $x^2+y^2-6x-4y-11=0$ from $P(1,8)$ touch the circle at $A$ and $B$. Then the centre of the circle passing through $P, A$ and $B$ is

Let $P$ and $Q$ be the inverse points with respect to the circle $S \equiv x^2+y^2-4x-6y+k=0$ and $C$ be the centre of the circle $S=0$ such that $CP \cdot CQ=4$. If $P=(1,2)$ and $Q=(a, b)$,then $2a=$

In a triangle,two vertices are $(2, 3)$ and $(4, 0)$,and its circumcentre is $(2, z)$ for some real number $z$. The circumradius is

If one end of the diameter of $x^2+y^2-2x-6y-15=0$ is $(4,1)$,then the coordinates of the other end are:

Let $\alpha$ be an integer multiple of $8$. If $S$ is the set of all possible values of $\alpha$ such that the line $6 x + 8 y + \alpha = 0$ intersects the circle $x^2 + y^2 - 4 x - 6 y + 9 = 0$ at two distinct points,then the number of elements in $S$ is

The straight line $(x - 2) + (y + 3) = 0$ cuts the circle $(x - 2)^2 + (y - 3)^2 = 11$ at