# Circle A has a radius of 2  and a center of (3 ,1 ). Circle B has a radius of 6  and a center of (8 ,5 ). If circle B is translated by <-4 ,3 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Feb 9, 2018

Since color(blue)(R_A + R_B (8)) color(green)(>) color(red)(vec(O_AO_B) (5.83) circles A & B overlap

#### Explanation:

${O}_{A} \left(\begin{matrix}3 \\ 1\end{matrix}\right) , {R}_{A} = 2$, ${O}_{B} \left(\begin{matrix}8 \\ 5\end{matrix}\right) , {R}_{B} = 6$

${O}_{B}$ translated by (-4,3)

New ${O}_{B} = \left(\begin{matrix}3 - 5 \\ 1 + 3\end{matrix}\right) \implies \left(\begin{matrix}- 2 \\ 4\end{matrix}\right)$

$\vec{{O}_{A} {O}_{B}} = s q r \left({\left(- 2 - 3\right)}^{2} + {\left(4 - 1\right)}^{2}\right) = \sqrt{34} \approx 5.83$

Sum of radii ${R}_{A} + {R}_{B} = 2 + 8 = 8$

Since ${R}_{A} + {R}_{B} \left(8\right) > \vec{{O}_{A} {O}_{B}} \left(5.83\right)$ the circles overlap