# Circle A has a radius of 2  and a center of (6 ,5 ). Circle B has a radius of 3  and a center of (2 ,4 ). If circle B is translated by <1 ,3 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Oct 10, 2017

Circle B overlaps circle A after translation.

#### Explanation:

If circle B is translated by <$1 , 3$>, the center will be ($2 + 1 , 4 + 3$)=($3 , 7$).
Let circle B' has a radius of $3$ and a center of ($3 , 7$).

The distance $d$ between the center of circle A and that of circle B' is:
$d = \sqrt{{\left(3 - 6\right)}^{2} + {\left(7 - 5\right)}^{2}} = \sqrt{13}$

Let ${r}_{a}$ and ${r}_{b}$ to the radius of circle A and circle B(and B') respectively. ${r}_{a} = 2 , {r}_{b} = 3$.

This satisfies the inequation:
$\left\mid {r}_{a} - {r}_{b} \right\mid < d < {r}_{a} + {r}_{b}$

Therefore circle A and circle B' (translated circle B) do neither circumscribe nor inscribe. They overlap.

The figure is cited from http://examist.jp/mathematics/figure-circle/two-circle/ (Japanese)