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

Question

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

Avery Allen · Accepted Answer

The circles do not overlap and are separated by a minimum distance of #2# units.

The length of the line segment joining the centers of the two circles is:
#color(white)("XXX")d=sqrt((3-(-1))^2+(-2-1)^2) =5#

Circle A covers #2# units of this line segment and
circle B covers #1# unit of this line segment,

leaving #5-(2+1)= 3# units of the line segment not covered by the circles.

Mason Adams · Answer

undefined Yes, the circles overlap. The smallest distance between the circles occurs at the point where the line segment connecting the centers of the circles intersects both circles. This distance can be calculated using the distance formula between two points. The distance between the centers of the circles is √((3 - (-1))^2 + (-2 - 1)^2) = √((4)^2 + (-3)^2) = √(16 + 9) = √25 = 5 units. Subtracting the sum of the radii from this distance gives us the smallest distance between the circles: 5 - (2 + 1) = 5 - 3 = 2 units. Therefore, the smallest distance between the circles is 2 units.