Circle A has a center at #(5 ,3 )# and a radius of #1 #. Circle B has a center at #(0 ,-5 )# 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 #(5   ,3   )# and a radius of #1 #.  Circle B has a center at #(0   ,-5    )# and a radius of #2 #.  Do the circles overlap?  If not, what is the smallest distance between them?

Violet Aldrich · Accepted Answer

Circles do not overlap each other. The smallest distance between the circles is #6.434#.

Distance between two centers #(5,3)# and #(0,−5)# is given by #sqrt((-5-3)^2+(0-5)^2)# or #sqrt(8^2+5^2# or #sqrt89=9.434#. As sum of the radii of two circles is #1+2=3# and is less than #9.434# which is distance between their centers, circles do not overlap each other. The smallest distance between the circles is #9.434-3=6.434#.

Carson Arnold · Answer

undefined The distance between the centers of Circle A and Circle B can be calculated using the distance formula:

$$ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Substituting the given coordinates, we find the distance between the centers of Circle A and Circle B:

$$ \text{Distance} = \sqrt{(0 - 5)^2 + (-5 - 3)^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.434 $$

Since the sum of the radii of Circle A and Circle B ($1 + 2 = 3$) is greater than the distance between their centers ($\sqrt{89}$), 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:

$$ \text{Smallest distance} = \text{Distance} - \text{Sum of radii} = \sqrt{89} - 3 \approx 6.434 $$