# Circle A has a radius of 2  and a center of (8 ,6 ). Circle B has a radius of 4  and a center of (2 ,3 ). If circle B is translated by <-1 ,2 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jul 23, 2017

The circles do not overlap and the distance is $= 1.07$

#### Explanation:

The coordinates of the center of circle $B '$ after translation is

${C}_{B} ' = \left(2 , 3\right) + < - 1 , 2 > = \left(1 , 5\right)$

The distance between the centers is

${C}_{A} {C}_{B} ' = \sqrt{{\left(1 - 8\right)}^{2} + {\left(5 - 6\right)}^{2}} = \sqrt{49 + 1} = \sqrt{50} = 7.07$

The sum of the radii is

${r}_{A} + {r}_{B} = 2 + 4 = 6$

As

${C}_{A} {C}_{B} ' > \left({r}_{A} + {r}_{B}\right)$, the circles do not overlap

The shortest distance is $= 7.07 - 6 = 1.07$

graph{((x-8)^2+(y-6)^2-4)((x-1)^2+(y-5)^2-16)=0 [-8.66, 16.65, -2.97, 9.69]}