The circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$ bisects the circumference of the circle ${x^2} + {y^2} + 2g'x + 2f'y + c' = 0$, if

The number of common tangents, to the circles $x^2+y^2-18 x-15 y+131=0$ and $x^2+y^2-6 x-6 y-7=0$, is :

The circles $x^2 + y^2 + 2x -2y + 1 = 0$ and $x^2 + y^2 -2x -2y + 1 = 0$ touch each  other :-

Two orthogonal circles are such that area of one is twice the area of other. If radius of smaller circle is $r$, then distance between their centers will be -

Two circles of radii $4$ cms $\&\,\, 1\,\, cm$ touch each other externally and $\theta$ is the angle contained by their direct common tangents. Then $sin \theta =$

If two circles ${(x - 1)^2} + {(y - 3)^2} = {r^2}$ and ${x^2} + {y^2} - 8x + 2y + 8 = 0$ intersect in two distinct points, then