# Circle A has a radius of 3  and a center of (8 ,7 ). Circle B has a radius of 2  and a center of (6 ,1 ). If circle B is translated by <2 ,7 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Feb 9, 2018

Circles Overlap

#### Explanation:

Given : ${O}_{A} \left(\begin{matrix}8 \\ 7\end{matrix}\right) , {R}_{A} = 3 , {O}_{B} \left(\begin{matrix}6 \\ 1\end{matrix}\right) , {R}_{B} = 2$

${O}_{B}$ translated by $\left(2 , 7\right)$

New ${O}_{B} \to \left(\begin{matrix}6 + 2 \\ 1 + 7\end{matrix}\right) \implies \left(\begin{matrix}8 \\ 8\end{matrix}\right)$

$\vec{{O}_{A} {O}_{B}} = \sqrt{{\left(8 - 8\right)}^{2} + {\left(8 - 7\right)}^{2}} = 1$

Sum of Radii ${R}_{A} + {R}_{B} = 3 + 2 = 5$

Since #R_A + R_B > vec(O_AO_B), both the circles overlap