Circle A has a radius of $3$ $3$ and a center of $(5, 4)$ $(5 ,4 )$ . Circle B has a radius of $1$ $1$ and a center of $(7, 2)$ $(7 ,2 )$ . If circle B is translated by $< 3, - 5 >$ $<3 ,-5 >$ , 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 $3$ $3$ and a center of $(5, 4)$ $(5 ,4 )$ . Circle B has a radius of $1$ $1$ and a center of $(7, 2)$ $(7 ,2 )$ . If circle B is translated by $< 3, - 5 >$ $<3 ,-5 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Answer 1 · 2018-02-16T15:18:15

$no overlap, \approx 4.602$ $"no overlap ",~~4.602$

Explanation:

$what we have to do here is compare the distance (d)$ $"what we have to do here is "color(blue)"compare"" the distance (d)"$
$to the sum of the radii$ $"to the "color(blue)"sum of the radii"$

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

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

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

$under the translation < 3, - 5 >$ $"under the translation "<3,-5>$

$(7, 2) \to (7 + 3, 2 - 5) \to (10, - 3) \leftarrow new centre of B$ $(7,2)to(7+3,2-5)to(10,-3)larrcolor(red)"new centre of B"$

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

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

$let (x_{1}, y_{1}) = (5, 4) and (x_{2}, y_{2}) = (10, - 3)$ $"let "(x_1,y_1)=(5,4)" and "(x_2,y_2)=(10,-3)$

$d = \sqrt{{(10 - 5)}^{2} + {(- 3 - 4)}^{2}} = \sqrt{25 + 49} \approx 8.062$ $d=sqrt((10-5)^2+(-3-4)^2)=sqrt(25+49)~~8.062$

$sum of radii = 3 + 1 = 4$ $"sum of radii "=3+1=4$

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

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

$\times \times \times \times \times = 8.062 - 4 = 4.062$ $color(white)(xxxxxxxxxx)=8.062-4=4.062$
graph{((x-5)^2+(y-4)^2-9)((x-10)^2+(y+3)^2-1)=0 [-20, 20, -10, 10]}

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Circle A has a radius of $3$ $3$ and a center of $(5, 4)$ $(5 ,4 )$ . Circle B has a radius of $1$ $1$ and a center of $(7, 2)$ $(7 ,2 )$ . If circle B is translated by $< 3, - 5 >$ $<3 ,-5 >$ , 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 33 and a center of (5,4)(5 ,4 ). Circle B has a radius of 11 and a center of (7,2)(7 ,2 ). If circle B is translated by <3,−5><3 ,-5 >, 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 $3$ $3$ and a center of $(5, 4)$ $(5 ,4 )$ . Circle B has a radius of $1$ $1$ and a center of $(7, 2)$ $(7 ,2 )$ . If circle B is translated by $< 3, - 5 >$ $<3 ,-5 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?