So my question goes like this "For the circles $C_1$: $x^2+y^2-10x+16y+89-r^2=0$ and $C_2$ $x^2+y^2+6x-14y+42=0$, the number of integral values of $r$ for which the circles are intersecting are?"
I do not have a doubt in the solving itself, but the problem is that in a general equation of a circle $(x-a)^2 + (y-b)^2 = r^2$, $r$ is always positive, but here in circle $C_1$ (simplifying, we get $(x-5)^2+(y+8)^2=r^2$), they have taken the value of radius as $|r|$ and found the number of integral values of $r$ ($13<|r|<21$, so $14$ values for $r$). Is this the correct way? If yes, then when do we take radius simply $r$, and when do we take it as $|r|$?
That's correct. Radii of circles drawn are $(13,4).$ Extreme radii at $(A,B)$ are $(13,13+8=21)$ excluded in a count as making tangential contact (black).
The intersecting (like blue) circles have radii $(14,15,16,17,18,19,20),$ seven non-tangential integer radii in all.