about to only mathematics

Explanation

We have circle`(x-1)^(2)+(y-3)^(2)=r^(2)` having centre `C_(1)(1,3)` and radius `r_(1)=r` and circle `x^(2)+^(2)-8x+2y+8=0` having centre `C_(2)(4,-1)` and `r_(2)=sqrt(4^(2)+(-1)^(2)-8)=3`.
Circles intersect in two distinct points if
`|r_(1)-r_(2)|ltC_(1)C_(2)ltr_(1)+r_(2)`
`:. |r-3| lt 5ltr+3`
`implies |r-3| lt5` and `rgt2`
`implies -5 lt r-3lt5` and `rgt2`
`implies -2 lt rlt8` and `rgt2`
`implies 2ltrlt8`