# Circle A has a center at #(1 ,-2 )# and a radius of #3 #. Circle B has a center at #(-4 ,-8 )# and a radius of #2 #. Do the circles overlap? If not, what is the smallest distance between them?

##### 1 Answer

Smallest distance between the two circles is

#### Explanation:

Let's compute the distance between the two centers of the circle.

This can be done with Pythagorean theorem:

#d^2 = d_x^2 + d_y^2#

where

In your case, you have

#d^2 = (1 -(-4))^2 + (-2 - (-8))^2 = 25 + 36 = 61#

#=> d = sqrt(61) ~~ 7.81#

However,

To compute the distance between the circles, we also need to take the radius

#"smallest distance" = d - r_1 - r_2 = d - 3 - 2 = sqrt(61) - 5 ~~ 2.81#

Thus, the smallest distance between these two circles is

I will insert a graph of the two circles with the line that passes through both centers.

The smallest distance is the length of this line between the two circles.

graph{((x-1)^2 + (y+2)^2 - 9)((x+4)^2 + (y+8)^2 - 4)(y - (6/5 x - 16/5)) = 0 [-13.92, 11.4, -10.73, 1.93]}