# Circle A has a radius of 3  and a center of (5 ,9 ). Circle B has a radius of 4  and a center of (1 ,2 ). If circle B is translated by <3 ,-1 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Feb 9, 2018

circles do not overlap

Minimum distance between circles A & B is $8.06 - 7 = \textcolor{red}{1.06}$

#### Explanation:

${R}_{A} = 3 , {O}_{A} \left(\begin{matrix}5 \\ 9\end{matrix}\right) , {R}_{B} = 4 , {O}_{B} \left(\begin{matrix}1 \\ 2\end{matrix}\right)$.

Center of circle B translated by $\left(\begin{matrix}3 \\ - 1\end{matrix}\right)$

Sum of radii ${R}_{A} + {R}_{B} = 3 + 4 = \textcolor{p u r p \le}{7}$

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

$\vec{{O}_{A} {O}_{B}} = \sqrt{{\left(5 - 4\right)}^{2} + {\left(9 - 1\right)}^{2}} = \sqrt{65} \approx \textcolor{g r e e n}{8.06}$

$\vec{{O}_{A} {O}_{B}} > {R}_{A} + {R}_{B}$

Hence the circles do not overlap

Minimum distance between circles A & B is $8.06 - 7 = \textcolor{red}{1.06}$

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