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

Question

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

Wyatt Anderson · Accepted Answer

The circles overlap

Circle A's center: #(-2.-4)# Circle B's center: #(-3,2)# Distance between centers: #sqrt((-3-(-2))^2+(2-(-4))^2)=sqrt(37)~~6.08# Along the line segment joining the two centers: circle A covers 2 units and circle B covers 5 units for a total of 7 units covered by the circles. But the line segment joining the two circles only has a length of #6.08# units. So the circles must overlap (by about #0.92# units)

James Atkins · Answer

undefined The circles do not overlap. The distance between their centers can be calculated using the distance formula, which is the square root of the sum of the squares of the differences in their coordinates. The distance between the centers of Circle A and Circle B is approximately $ 6.403 $ units.

The sum of their radii is $ 2 + 5 = 7 $ units. Since the distance between their centers is greater than the sum of their radii, the circles do not overlap.

The smallest distance between the circles is the difference between the distance between their centers and the sum of their radii. So, the smallest distance between the circles is approximately $ 6.403 - 7 = -0.597 $ units. However, distances are non-negative, so we consider the absolute value. Thus, the smallest distance between the circles is approximately $ 0.597 $ units.