If $P$ and $Q$ are the points of intersection of the circles ${x^2} + {y^2} + 3x + 7y + 2p - 5 = 0$ and ${x^2} + {y^2} + 2x + 2y - {p^2} = 0$ then there is a circle passing through $P, Q$ and $(1, 1)$ for:

The number of common tangents to the circles ${x^2} + {y^2} = 4$ and ${x^2} + {y^2} - 6x - 8y = 24$ is

In the co-axial system of circle ${x^2} + {y^2} + 2gx + c = 0$, where $g$ is a parameter, if $c > 0$ then the circles are

The number of common tangents to the circles ${x^2} + {y^2} - 4x - 6y - 12 = 0$ and ${x^2} + {y^2} + 6x + 18y + 26 = 0$ is

The circles ${x^2} + {y^2} + 4x + 6y + 3 = 0$ and $2({x^2} + {y^2}) + 6x + 4y + C = 0$ will cut orthogonally, if $C$ equals

Let $C_1, C_2$ be two circles touching each other externally at the point $A$ and let $A B$ be the diameter of circle $C_1$. Draw a secant $B A_3$ to circle $C_2$, intersecting circle $C_1$ at a point $A_1(\neq A)$, and circle $C_2$ at points $A_2$ and $A_3$. If $B A_1=2, B A_2=3$ and $B A_3=4$, then the radii of circles $C_1$ and $C_2$ are respectively