# Circle A has a center at (3 ,7 ) and a radius of 2 . Circle B has a center at (4 ,-1 ) and a radius of 6 . Do the circles overlap? If not, what is the smallest distance between them?

Feb 18, 2016

The circles do not overlap;
there is a minimum distance of (approximately) $0.06$ between them

#### Explanation:

The distance between the centers of A and B is
$\textcolor{w h i t e}{\text{XXX}} \sqrt{{\left(3 - 4\right)}^{2} + {\left(7 - \left(- 0\right)\right)}^{2}} = \sqrt{65} \approx 8.06$

Considering the line segment which joins the two centers
$\textcolor{w h i t e}{\text{XXX}}$Circle A covers 2 units of this line segment
$\textcolor{w h i t e}{\text{XXX}}$Circle B covers 6 units of this line segment.

Together the two circles cover $\left(2 + 6\right) = 8$ units of the line segment which is $8.06$ long; leaving a distance of $0.06$ between the two circles.