The number of common tangents to two circles ${x^2} + {y^2} = 4$ and ${x^2} - {y^2} - 8x + 12 = 0$ is

If the centre of a circle which passing through the points of intersection of the circles ${x^2} + {y^2} - 6x + 2y + 4 = 0$and ${x^2} + {y^2} + 2x - 4y - 6 = 0$ is on the line $y = x$, then the equation of the circle is

Figure shows $\Delta  ABC$ with $AB = 3, AC = 4$  &  $BC = 5$. Three circles $S_1, S_2$  &  $S_3$ have their centres on $A, B  $ &  $C$ respectively and they externally touches each other. The sum of areas of three circles is

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

The value of $\lambda $, for which the circle ${x^2} + {y^2} + 2\lambda x + 6y + 1 = 0$, intersects the circle ${x^2} + {y^2} + 4x + 2y = 0$ orthogonally is

Let a circle $C_1 \equiv  x^2 + y^2 - 4x + 6y + 1 = 0$ and circle $C_2$ is such that it's centre is image of centre of $C_1$ about $x-$axis and radius of $C_2$ is equal to radius of $C_1$, then area of $C_1$ which is not common with $C_2$ is -