Circle A has a center at (3 ,5 ) and a radius of 1 . Circle B has a center at (-1 ,1 ) and a radius of 4 . Do the circles overlap? If not, what is the smallest distance between them?

Feb 2, 2016

The circles do not overlap.
The minimum distance between them is $4 \sqrt{2} - 5 \approx 0.66$

Explanation:

The distance between the centers of the two circles is
$\textcolor{w h i t e}{\text{XXX}} \sqrt{{\left(3 - \left(- 1\right)\right)}^{2} + {\left(5 - 1\right)}^{2}} = 4 \sqrt{2} \approx 5.66$

Consider a line segment joining the two centers.
The distance from A's center to A's circumference is $1$.
The distance from B's center to B's circumference is $4$.
The sum of the radial distances is $1 + 4 = 5$ which is less than the length of the line segment $4 \sqrt{2} \approx 5.66$
by $5.66 - 5 = 0.66$ (units).