If two circles `(x-1)^(2) +( y-3)^(2) =r^(2)` and `x^(2) + y^(2) -8x+2y +8 =0` intersect in two distinct points, then `:`

Options

A. `2 lt r lt 8`
B. `r lt 2`
C. `r=2`
D. `r gt 2`

Answer

A

Explanation

If d is the distance between the centres of two circles of radii`r_(1) and r_(2)`, then they intersect in two distinct points iff
`|r_(1)-r_(2)| lt d lt r_(1)+r_(2)`
Here , radii of two circles are r and 3 and distance between the centres is 5. So, the circles will intersect in two distinct points iff.
`|r-3|lt5ltr+3rArr -2 lt r lt 8 and r gt 2 rArr 2 lt r lt 8`