If the polar of a point $P$ with respect to a circle of radius $r$ which touches the coordinate axes and lies in the first quadrant is $x+2 y=4 r$, then the point $P$ is

If the circles $x^2+y^2-2 x-2(3+\sqrt{7}) y+8+6 \sqrt{7}=0$ and $x^2+y^2-8 x-6 y+k^2=0, k \in \mathbf{Z}$, have exactly two common tangents, then the number of possible values of $k$ is

The circle $S=0$ cuts the circles

$C_1=x^2+y^2-8 x-2 y+16=0$ and $C_2=x^2+y^2-4 x-4 y-1=0$ orthogonally. If the common chord of $S=0$ and $C_1=0$ is $2 x+13 y-15=0$, then the centre of $S=0$ is

The equation of the circle passing through the points of intersection of the two orthogonal circles $S_1=x^2+y^2+k x-4 y-1=0$, $S_2=3 x^2+3 y^2-14 x+23 y-15=0$ and passing through the point $(-1,-1)$ is