# Circle A has a radius of 1  and a center at (2 ,3 ). Circle B has a radius of 3  and a center at (6 ,4 ). 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?

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Feb 9, 2018

Since ${R}_{A} + {R}_{B} < \vec{{O}_{A} {O}_{B}}$, circles do NOT overlap

Minimum distance between the two circles A & B is 5.1 - 4 = color(red)(1.1

#### Explanation:

Given : ${O}_{A} \left(2 , 3\right) , {R}_{A} = 1 , {O}_{B} \left(6 , 4\right) , {R}_{B} = 3$

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

New coordinates of ${O}_{B} \left(\begin{matrix}6 - 3 \\ 4 + 4\end{matrix}\right) \implies \left(\begin{matrix}3 \\ 8\end{matrix}\right)$

Now, $\vec{{O}_{A} {O}_{B}} = \sqrt{{\left(2 - 3\right)}^{2} + {\left(3 - 8\right)}^{2}} = \sqrt{26} \approx 5.1$

Sum of radii ${R}_{A} + {R}_{B} = 1 + 3 = 4$

Since ${R}_{A} + {R}_{B} < \vec{{O}_{A} {O}_{B}}$, circles do NOT overlap

Minimum distance between the two circles A & B is $5.1 - 4 = 1.1$

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