# Circle A has a radius of 5  and a center of (2 ,6 ). Circle B has a radius of 2  and a center of (4 ,3 ). If circle B is translated by <-3 ,1 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jul 15, 2017

The circles overlap.

#### Explanation:

The new center of circle $B$ is

$C {'}_{B} = \left(4 , 3\right) + < - 3 , 1 > = \left(1 , 4\right)$

The distance between the centers of the circles after translation is

${C}_{A} C {'}_{B} = \sqrt{{\left(1 - 2\right)}^{2} + {\left(4 - 6\right)}^{2}} = \sqrt{1 + 4} = \sqrt{5}$

The sum of the radii is

$= {r}_{A} + {r}_{B} ' = 5 + 2 = 7$

As ${C}_{A} {C}_{B} ' < {r}_{A} + {r}_{B}$, the circles overlap

graph{((x-2)^2+(y-6)^2-25)((x-1)^2+(y-4)^2-4)=0 [-10, 10, -5, 5]}