The circles $${x^2} + {y^2} - 10x + 16 = 0$$     and $${x^2} + {y^2} = {r^2}$$   intersect each other in two distinct points if-

Solution :
Centres and radii of two circles are $${C_1}\left( {5,\,0} \right);\,3 = {r_1},$$     and $${C_2}\left( {0,\,0} \right);\,r = {r_2}$$
As circles intersect each other in two distinct points,
$$\eqalign{ & \left| {{r_1} - {r_2}} \right| < {C_1}{C_2} < {r_1} + {r_2} \cr & \Rightarrow \left| {r - 3} \right| < 5 < r + 3 \cr & \Rightarrow 2 < r < 8 \cr} $$