Circle A has a center at #(2 ,3 )# and a radius of #1 #. Circle B has a center at #(0 ,-2 )# and a radius of #4 #. Do the circles overlap? If not, what is the smallest distance between them?

Question

Circle A has a center at #(2   ,3   )# and a radius of #1 #.  Circle B has a center at #(0   ,-2    )# and a radius of #4 #.  Do the circles overlap?  If not, what is the smallest distance between them?

Jayden Jackson · Accepted Answer

no overlap , d ≈ 0.385

First step is to calculate the distance between the centres using the #color(blue)" distance formula " # # d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2) # where # (x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points "# let # (x_1,y_1)=(2,3)" and " (x_2,y_2)= (0,-2) # #rArr d = sqrt((0-2)^2+(-2-3)^2) = sqrt(4+25) ≈ 5.385 # radius of A + radius of B = 1 + 4 = 5 since : radius of A + radius of B < distance between centres there is no overlap. distance between circles ≈ 5.385 - 5 ≈ 0.385

Avery Allen · Answer

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

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

Substituting the given coordinates:

$$d = \sqrt{(0 - 2)^2 + (-2 - 3)^2}$$

$$d = \sqrt{(-2)^2 + (-5)^2}$$

$$d = \sqrt{4 + 25}$$

$$d = \sqrt{29}$$

Since the distance between the centers of the circles ($\sqrt{29}$) is greater than the sum of their radii (1 + 4 = 5), 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:

$$ \text{Smallest distance} = \sqrt{29} - (1 + 4)$$

$$ \text{Smallest distance} = \sqrt{29} - 5$$