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

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

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Answer 1 · 2018-03-01T12:01:48

$no overlap \approx 0.71 units$ $"no overlap "~~0.71" units"$

Explanation:

$What we have to do here is to compare the$ $"What we have to do here is to "color(blue)"compare ""the"$
$distance (d) between the centres with the sum of radii$ $"distance (d) between the centres with the "color(blue)"sum of radii"$

$∙ if sum of radii > d then circles overlap$ $• " if sum of radii">d" then circles overlap"$

$∙ if sum of radii < d then no overlap$ $• " if sum of radii"< d" then no overlap"$

$Before calculating d we require to find the centre of B$ $"Before calculating d we require to find the centre of B"$
$under the given translation$ $"under the given translation"$

$under the translation < - 2, 4 >$ $"under the translation "<-2,4>$

$(8, 3) \to (8 - 2, 3 + 4) \to (6, 7) \leftarrow new centre of B$ $(8,3)to(8-2,3+4)to(6,7)larrcolor(red)"new centre of B"$

$to calculate d use the distance formula$ $"to calculate d use the "color(blue)"distance formula"$

$∙ x d = \sqrt{{(x_{2} - x_{1})}_{2}^{2} + {(y_{2} - y_{1})}_{2}^{2}}$ $•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)$

$let (x_{1}, y_{1}) = (3, 1) and (x_{2}, y_{2}) = (6, 7)$ $"let "(x_1,y_1)=(3,1)" and "(x_2,y_2)=(6,7)$

$d = \sqrt{{(6 - 3)}^{2} + {(7 - 1)}^{2}} = \sqrt{9 + 36} = \sqrt{45} \approx 6.71$ $d=sqrt((6-3)^2+(7-1)^2)=sqrt(9+36)=sqrt45~~6.71$

$sum of radii = 2 + 4 = 6$ $"sum of radii "=2+4=6$

$since sum of radii < d then no overlap$ $"Since sum of radii"< d" then no overlap"$

$min. distance = d - sum of radii$ $"min. distance "=d-" sum of radii"$

$min. distance = 6.71 - 6 = 0.71$ $color(white)("min. distance ")=6.71-6=0.71$
graph{((x-3)^2+(y-1)^2-4)((x-6)^2+(y-7)^2-16)=0 [-20, 20, -10, 10]}

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

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Explanation:

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Circle A has a radius of 22 and a center at (3,1)(3 ,1 ). Circle B has a radius of 44 and a center at (8,3)(8 ,3 ). If circle B is translated by <−2,4><-2 ,4 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Explanation:

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