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 :

$P, Q$ and $R$ are the centres and ${r_1},\,\,{r_2},\,\,{r_3}$ are the radii respectively of three co-axial circles, then $QRr_1^2 + RP\,r_2^2 + PQr_3^2$ is equal to

Suppose $S_1$ and $S_2$ are two unequal circles, $A B$ and $C D$ are the direct common tangents to these circles. A transverse common tangent $P Q$ cuts $A B$ in $R$ and $C D$ in $S$. If $A B=10$, then $R S$ is

The locus of centre of a circle passing through $(a, b)$ and cuts orthogonally to circle ${x^2} + {y^2} = {p^2}$, is

The set of all real values of $\lambda $ for which exactly two common tangents can be drawn to the circles $x^2 + y^2 – 4x – 4y+ 6\, = 0$ and $x^2 + y^2 – 10x – 10y + \lambda \, = 0$ is the interval: