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

Question

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

Evelyn Arboleda · Accepted Answer

smallest distance = #1.40312# Circle #A# center is located at #c_A=(-4,-1)# Circle #B# center is located at #c_B=(1,3)# Their distance is #d_{AB} = norm(c_A-c_B) = 6.40312#. If they were tangents their center distance will be #r_A+r_B#. Where #r_A,r_B# are their respective radius. In the present case we have #d_{AB} > r_A+r_B# so their smallest distance is given by #d_{AB} -( r_A+r_B) = 1.40312#

Sadie Adams · Answer

undefined The distance between the centers of Circle A and Circle B can be calculated using the distance formula. It is $ \sqrt{(1 - (-4))^2 + (3 - (-1))^2} = \sqrt{5^2 + 4^2} = \sqrt{25 + 16} = \sqrt{41} \approx 6.4 $ units. Since the sum of the radii of the two circles is $ 3 + 2 = 5 $ units, which is less than the distance between their centers, the circles do not overlap. The smallest distance between them is the difference between the distance between their centers and the sum of their radii, which is $ \sqrt{41} - 5 $ units. Therefore, the smallest distance between the circles is approximately $ 6.4 - 5 = 1.4 $ units.