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

Circles ${x^2} + {y^2} - 2x - 4y = 0$ and ${x^2} + {y^2} - 8y - 4 = 0$

The point $(2, 3)$ is a limiting point of a coaxial system of circles of which ${x^2} + {y^2} = 9$ is a member. The co-ordinates of the other limiting point is given by

A circle $C_1$ of radius $2$ touches both $x$ -axis and $y$ -axis. Another circle $C_2$ whose radius is greater than $2$ touches circle $C_1$ and both the axes. Then the radius of  circle $C_2$ is-

Let the equation $x^{2}+y^{2}+p x+(1-p) y+5=0$ represent circles of varying radius $\mathrm{r} \in(0,5]$. Then the number of elements in the set $S=\left\{q: q=p^{2}\right.$ and $\mathrm{q}$ is an integer $\}$ is ..... .

If a variable line, $3x + 4y -\lambda  = 0$ is such that the two circles $x^2 + y^2 -2x -2y + 1 = 0$ and $x^2 + y^2 -18x -2y + 78 = 0$ are on its opposite sides, then the set of all values of $\lambda $ is the interval