If the circle $x^2+y^2+8x-4y+c=0$ touches the circle $x^2+y^2+2x+4y-11=0$ externally and cuts the circle $x^2+y^2-6x+8y+k=0$ orthogonally,then $k$ is equal to

Any circle passes through the point of intersection of the lines $x + \sqrt{3}y = 1$ and $\sqrt{3}x - y = 2$. If it intersects these lines at points $P$ and $Q$,then the angle subtended by the arc $PQ$ at its centre is ............ $^o$.

The circles $x^2 + y^2 + 4x + d = 0$ and $x^2 + y^2 + 4fy + d = 0$ touch each other if:

If the angle between the circles $x^2+y^2+2x-4y+1=0$ and $x^2+y^2-4x-2y+c=0$ is $\frac{\pi}{4}$,then $c=$

The equation of the tangent at the point $(0, 3)$ on the circle which cuts the circles $x^2 + y^2 - 2 x + 6 y = 0$,$x^2 + y^2 - 4 x - 2 y + 6 = 0$ and $x^2 + y^2 - 12 x + 2 y + 3 = 0$ orthogonally is

$A$ circle $S$ passes through the points of intersection of the circles $x^2+y^2-2x-3=0$ and $x^2+y^2-2y=0$. If $x+y+1=0$ is a tangent to the circle $S$,then the equation of $S$ is