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

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Answer 1 · 2017-10-25T19:35:50

$circles touch externally$ $"circles touch externally"$

Explanation:

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

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

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

We require to find the 'new' centre of B under the given translation which does not change the shape of the circle only it's position.

$under a translation < 2, - 1 >$ $"under a translation "<2,-1>$

$(3, 2) \to (3 + 2, 2 - 1) \to (5, 1) \leftarrow new centre of B$ $(3,2)to(3+2,2-1)to(5,1)larrcolor(red)" new centre of B"$

Note that the x-coordinate of the centres of both circles is 5 indicating that they lie on the vertical line x = 5
Hence d is the difference in the y-coordinates.

$\Rightarrow d = 7 - 1 = 6$ $rArrd=7-1=6$

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

$since sum of radii = d = 6$ $"since sum of radii "=d=6$

$then the circles touch externally$ $"then the circles touch externally"$
graph{((x-5)^2+(y-7)^2-4)((x-5)^2+(y-1)^2-16)=0 [-20, 20, -10, 10]}

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Circle A has a radius of $2$ $2$ and a center of $(5, 7)$ $(5 ,7 )$ . Circle B has a radius of $4$ $4$ and a center of $(3, 2)$ $(3 ,2 )$ . If circle B is translated by $< 2, - 1 >$ $<2 ,-1 >$ , 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 of (5,7)(5 ,7 ). Circle B has a radius of 44 and a center of (3,2)(3 ,2 ). If circle B is translated by <2,−1><2 ,-1 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

1 Answer

Explanation:

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