Question

Circle A has a center at `(-6 ,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?

Answer 1

Yes, they overlap.

Explanation:

As the radii of the circles sum to `14`, the centers of circle A and circle B need to be at greater than `14` units apart for the circles to not overlap at any point.

To see why this is the case, construct the line segment from the center of A to the center of B. All points on the segment with distance `<=9` from the center of A are inside or on A. Similarly, all points with distances `<=5` from the center of B are inside or on B. For these two sections of the segment to not intersect, the line segment would need to be greater than `14` units in length.

The distance between the center of circle A and the center of circle B is `sqrt((1-(-6))^2+(-5-4)^2)=sqrt(49+81)=sqrt(130)~~11.4`

As `11.4 < 14`, the circles overlap.