# Circle A has a radius of 3  and a center of (5 ,4 ). Circle B has a radius of 4  and a center of (7 ,2 ). If circle B is translated by <3 ,2 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jul 6, 2017

The circles overlap

#### Explanation:

Let the center of the circle $A$ be ${O}_{A} = \left(5 , 4\right)$

and the center of the circle $B$ be ${O}_{B} = \left(7 , 2\right)$

The center after translation is

${O}_{B} ' = \left(7 , 2\right) + < 3 , 2 > = \left(10 , 4\right)$

The distance between the centers is

${O}_{A} {O}_{B} ' = \sqrt{{\left(10 - 5\right)}^{2} + {\left(4 - 4\right)}^{2}}$

$= 5$

The sum of the radii is

${r}_{A} + {r}_{B} = 3 + 4 = 7$

As ,

${O}_{A} {O}_{B} ' < \left({r}_{A} + {r}_{B}\right)$

The circles overlap

graph{((x-5)^2+(y-4)^2-9)((x-7)^2+(y-2)^2-16)((x-10)^2+(y-4)^2-16)=0 [-6, 16.5, -2.185, 9.065]}