If the circles $x^2+y^2-16x-20y+164=r^2$ and

$(x-4{)}^2+(y-7{)}^2=36$ intersect in two distinct points, then

 

The coordinates of the centres of two circles are $C_1(8,10)$and $C_2(4,7)$ respectively. Their radii are $r_1=r$ and $r_2=6$

respectively. It the two circles intersect at two distinct points, then

$\begin{array}{l}\left|r_1-r_2\right|<C_1{C}_2<r_1+r_2\\ \Rightarrow |r-6|<5<r+6\\ \Rightarrow |r-6|<5\text{ and }5<r+6\Rightarrow |r-6|<5\Rightarrow 1<r<11\end{array}$