# Circle A has a radius of 1  and a center of (5 ,2 ). Circle B has a radius of 2  and a center of (4 ,5 ). If circle B is translated by <-3 ,4 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jul 13, 2016

After the translation ${C}_{B} = \left(4 - 3 , 5 + 4\right) = \left(1 , 9\right)$

#### Explanation:

The distance between ${C}_{A} \mathmr{and} {C}_{B}$ can be calculated by the good old Pythagoras:
${\left({C}_{A} {C}_{B}\right)}^{2} = {\left(5 - 1\right)}^{2} + {\left(2 - 9\right)}^{2} = 16 + 49 = 65$
$\to {C}_{A} {C}_{B} = \sqrt{65} \approx 8.06$

Since the radii of the circles add up to $3$, they don't overlap, and the minimum distance between points on the circles will be:
$d = \sqrt{65} - 3 \approx 5.06$