# Circle A has a radius of 3  and a center of (2 ,7 ). Circle B has a radius of 2  and a center of (6 ,1 ). If circle B is translated by <2 ,7 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jun 28, 2017

The circles do not overlap and the minimum distance is $= 1.08$

#### Explanation:

Circle $A$, center ${O}_{A} = \left(2 , 7\right)$

The equation of circle $A$ is

${\left(x - 2\right)}^{2} + {\left(y - 7\right)}^{2} = 9$

Circle $B$, center ${O}_{B} = \left(6 , 1\right)$

The equation of circle $B$ is

${\left(x - 6\right)}^{2} + {\left(y - 1\right)}^{2} = 4$

The center of circle $B '$ after translation is

$\left(6 , 1\right) + \left(2 , 7\right) = \left(8 , 8\right)$

Circle $B '$, center ${O}_{B} ' = \left(8 , 8\right)$

The equation of the circle after translation is

${\left(x - 8\right)}^{2} + {\left(y - 8\right)}^{2} = 4$

The distance ${O}_{A} {O}_{B} '$ is

$= \sqrt{{\left(8 - 2\right)}^{2} + {\left(8 - 7\right)}^{2}}$

$= \sqrt{36 + 1}$

$= \sqrt{37}$

$= 6.08$

This distance is greater than the sum of the radii

${O}_{A} {O}_{B} ' > {r}_{A} + {r}_{B} '$

So, the circles do not overlap and the minimum distance is

$= 6.08 - \left(2 + 3\right)$

$= 1.08$
graph{((x-2)^2+(y-7)^2-9)((x-6)^2+(y-1)^2-4)((x-8)^2+(y-8)^2-4)=0 [-7.28, 18.03, -1.57, 11.09]}