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

##### 1 Answer
Jul 21, 2017

The circles do not overlap. The minimum distance is $= 1.3$

#### Explanation:

The sum of the radii of the circles is

${r}_{A} + {r}_{B} = 1 + 4 = 5$

The center of circle $B$ after translation is

$= \left(5 , 3\right) + < - 2 , 5 \ge \left(3 , 8\right)$

The distance between the centers is

$d = \sqrt{{\left(3 - 1\right)}^{2} + {\left(8 - 2\right)}^{2}} = \sqrt{4 + 36} = \sqrt{40} = 6.3$

As

$d > {r}_{A} + {r}_{B}$

The circles do not overlap.

The minimum distance is

$= d - \left({r}_{A} + {r}_{B}\right) = 6.3 - 5 = 1.3$

graph{((x-1)^2+(y-2)^2-1)((x-3)^2+(y-8)^2-16)=0 [-12.58, 15.9, -3, 11.24]}