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

Feb 9, 2018

Two circles DO NOT OVERLAP

Minimum distance between both circles = 5.83 - 3 = color(brown)(2.83

#### Explanation:

${O}_{A} \left(\begin{matrix}2 \\ 4\end{matrix}\right) , {R}_{A} = 1 , {O}_{B} \left(\begin{matrix}4 \\ 5\end{matrix}\right) , {R}_{B} = 2$

Circle B translated by $\left(\begin{matrix}1 \\ 4\end{matrix}\right)$

Sum of Radii R_A + R_B = 1 + 2 = color(green)(3

New coordinates of ${O}_{B} = \left(\begin{matrix}4 + 1 \\ 5 + 4\end{matrix}\right) \implies \left(\begin{matrix}5 \\ 9\end{matrix}\right)$

Distance d = vec(O_AO_B) = sqrt((5-2)^2 + (9-4)^2) = sqrt34 ~~ color(blue)(5.83

${R}_{a} + {R}_{B} < \vec{{O}_{A} {O}_{B}}$. Hence the Two circles DO NOT OVERLAP

Minimum distance between both circles = 5.83 - 3 = color(brown)(2.83