# Circle A has a center at (1 ,-8 ) and a radius of 3 . Circle B has a center at (-2 ,-5 ) and a radius of 2 . Do the circles overlap? If not, what is the smallest distance between them?

Jun 4, 2016

The sum of the radii of the circles is $5$ $u n i t s$. If the distance between their centres is less than this they will overlap. If it is more, the distance between their closest points is the distance between the centres minus 5.

#### Explanation:

Distance between the centres of the circles:

$r = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}} = \sqrt{{\left(- 2 - 1\right)}^{2} + {\left(- 5 - - 8\right)}^{2}}$
$= \sqrt{{\left(- 3\right)}^{2} + {\left(3\right)}^{2}} = \sqrt{9 + 9} = \sqrt{18} \approx 4.2$

Since this is less than $5$ $u n i t s$, the circles will overlap.