The equation of director circle of the circle ${x^2} + {y^2} = {a^2},$ is 

The centre of the smallest circle touching the circles $x^2 + y^2- 2y - 3 = 0$ and $x^2+ y^2 - 8x - 18y + 93 = 0$ is :

If circles ${x^2} + {y^2} + 2ax + c = 0$and ${x^2} + {y^2} + 2by + c = 0$ touch each other, then 

Let the latus ractum of the parabola $y ^{2}=4 x$ be the common chord to the circles $C _{1}$ and $C _{2}$ each of them having radius $2 \sqrt{5}$. Then, the distance between the centres of the circles $C _{1}$ and $C _{2}$ is

Let $C_i \equiv  x^2 + y^2 = i^2 (i = 1,2,3)$ are three circles. If there are $4i$ points on circumference of circle $C_i$. If no three of all the points on three circles are collinear then number of triangles which can be formed using these points whose circumcentre does not lie on origin, is-

The equation of radical axis of the circles $2{x^2} + 2{y^2} - 7x = 0$ and ${x^2} + {y^2} - 4y - 7 = 0$ is