# Circle A has a center at (1 ,-4 ) and a radius of 2 . Circle B has a center at (9 ,3 ) and a radius of 5 . Do the circles overlap? If not what is the smallest distance between them?

Feb 16, 2016

The circles do not overlap;
they are separated by a minimum distance of $1.6$ units

#### Explanation:

The distance between the centers of the circles:
$\textcolor{w h i t e}{\text{XXX}} d = \sqrt{{\left(1 - 9\right)}^{2} + {\left(- 4 - 3\right)}^{2}} = \sqrt{113} \approx 10.6$

Circle A covers 2 (units) of the distance along the line segment from the center of A to the center of B.

Circle B covers 5 (units) of the distance along the line segment form the center of B to the center of A.

A distance of
$\textcolor{w h i t e}{\text{XXX}} 10.6 - \left(2 + 7\right) = 1.6$ along the line segment joining the two centers
is not covered by either circle.