# Circle A has a radius of 2  and a center at (3 ,1 ). Circle B has a radius of 4  and a center at (8 ,3 ). If circle B is translated by <-4 ,-1 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jan 13, 2018

They don't overlap. The minimum distance is $2 - \sqrt{2}$.

#### Explanation:

The new center of circle B is $\left(4 , 2\right)$.

The distance between the centers of the two circles is

$\sqrt{{\left(4 - 3\right)}^{2} + {\left(2 - 1\right)}^{2}} = \sqrt{2}$

Circle B is the larger circle and all points on the circle are 4 units from its center.

The radius of circle A is 2 and the center of A is $\sqrt{2}$ units from the center of Circle B so the farthest any point on A can be from the center of B is $2 + \sqrt{2}$. Since $2 + \sqrt{2} < 4$ no point on A overlaps any point on B.

The minimum distance between points on A and B is actually the radius of B, which is 4, minus the farthest any point on A can be from the center of B, which is $2 + \sqrt{2}$:

$4 - \left(2 + \sqrt{2}\right) = 2 - \sqrt{2}$