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 ,2 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Aug 18, 2017

The distance between the centers of the circles is $\sqrt{2}$, and since the radius of circle a is 5, the circles will definitely overlap.

Explanation:

The new center of circle B will be $\left(4 - 3 , 3 + 2\right) = \left(1 , 5\right)$.

We can now find the distance between the centres of the two circles:

$r = \sqrt{{\left({x}_{a} - {x}_{b}\right)}^{2} + {\left({y}_{a} - {y}_{b}\right)}^{2}}$

$= \sqrt{{\left({x}_{a} - {x}_{b}\right)}^{2} + {\left({y}_{a} - {y}_{b}\right)}^{2}}$

$= \sqrt{{\left(2 - 1\right)}^{2} + {\left(6 - 5\right)}^{2}} = \sqrt{2} = 1.414$

Given that the radius of circle a is 5, the circles will definitely overlap.