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

Jul 16, 2016

The two circles intersect.

#### Explanation:

The distance between the centers of the circles is given by
$\sqrt{{\left(2 - 1\right)}^{2} + {\left(4 - \left(- 5\right)\right)}^{2}} = \sqrt{{1}^{2} + {9}^{2}} = \sqrt{82}$

This is more than the difference of the two radii ($9 - 5 = 4$) and less than their sum ($9 + 5 = 14$). So, the two circles intersect.

Note : If the distance had been less than the difference of the two radii, thee smaller circle would have been entirely inside the larger one. On the other hand, had it been larger than the sum of the radii, the two circles would have been too far away to intersect. Neither is the case here.