# Circle A has a radius of 2  and a center of (2 ,7 ). Circle B has a radius of 6  and a center of (7 ,5 ). If circle B is translated by <-1 ,1 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Feb 7, 2018

circle B do not overlap.
Shortest distance is $3.764 - 2 = 1.764$

#### Explanation:

Circle A
Center $\left(2 , 7\right)$
radius $2$

Equation of circle A
${\left(x - 2\right)}^{2} + {\left(y - 7\right)}^{2} = {2}^{2}$

Circle B
Center $\left(7 , 5\right)$
radius $6$

Equation of circle B
${\left(x - 7\right)}^{2} + {\left(y - 5\right)}^{2} = {6}^{2}$

ranslation of B
<-1,1>
Center of B after translation $\left(\begin{matrix}7 - 1 \\ 5 + 1\end{matrix}\right)$
Center of B after translation $\left(6 , 6\right)$

Equation of circle B after translation
${\left(x - 6\right)}^{2} + {\left(y - 6\right)}^{2} = {6}^{2}$

Intersection of the circles
${\left(x - 2\right)}^{2} + {\left(y - 7\right)}^{2} = {2}^{2}$
and
${\left(x - 6\right)}^{2} + {\left(y - 6\right)}^{2} = {6}^{2}$
can be found out

Distance from center of Circle A to
center of Circle B is
$\left(2 , 7\right) - - - - \left(6 , 6\right)$
=sqrt((6-7)^2+(6-2)^2
=sqrt((1+4)
$= \sqrt{5}$
$= 2.236$

Radius of circle A is $2$
point on the Perimeter of the circle B is away from its center by 6
point on the Perimeter of the circle B is away from (2,7) is$6 - 2.236$
$= 3.764$
Hence, the two circles donot overlap.
Shortest distance is $3.764 - 2 = 1.764$

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