Circle A has a radius of $6$ $6$ and a center of $(2, 5)$ $(2 ,5 )$ . Circle B has a radius of $3$ $3$ and a center of $(1, 7)$ $(1 ,7 )$ . If circle B is translated by $< 3, 1 >$ $<3 ,1 >$ , 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 $6$ $6$ and a center of $(2, 5)$ $(2 ,5 )$ . Circle B has a radius of $3$ $3$ and a center of $(1, 7)$ $(1 ,7 )$ . If circle B is translated by $< 3, 1 >$ $<3 ,1 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Answer 1 · 2018-07-02T11:01:50

$circles overlap$ $"circles overlap"$

Explanation:

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

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

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

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

$under the translation < 3, 1 >$ $"under the translation "< 3,1>$

$(1, 7) \to (1 + 3, 7 + 1) \to (4, 8) \leftarrow new centre of B$ $(1,7)to(1+3,7+1)to(4,8)larrcolor(red)"new centre of B"$

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

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

$let (x_{1}, y_{1}) = (2, 5) and (x_{2}, y_{2}) = (4, 8)$ $"let "(x_1,y_1)=(2,5)" and "(x_2,y_2)=(4,8)$

$d = \sqrt{{(4 - 2)}^{2} + {(8 - 5)}^{2}} = \sqrt{4 + 9} = \sqrt{13} \approx 3.61$ $d=sqrt((4-2)^2+(8-5)^2)=sqrt(4+9)=sqrt13~~3.61$

$sum of radii = 6 + 3 = 9$ $"sum of radii "=6+3=9$

$since sum of radii > d then circles overlap$ $"Since sum of radii">d" then circles overlap"$
graph{((x-2)^2+(y-5)^2-36)((x-4)^2+(y-8)^2-9)=0 [-40, 40, -20, 20]}

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Circle A has a radius of $6$ $6$ and a center of $(2, 5)$ $(2 ,5 )$ . Circle B has a radius of $3$ $3$ and a center of $(1, 7)$ $(1 ,7 )$ . If circle B is translated by $< 3, 1 >$ $<3 ,1 >$ , 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 66 and a center of (2,5)(2 ,5 ). Circle B has a radius of 33 and a center of (1,7)(1 ,7 ). If circle B is translated by <3,1><3 ,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 $6$ $6$ and a center of $(2, 5)$ $(2 ,5 )$ . Circle B has a radius of $3$ $3$ and a center of $(1, 7)$ $(1 ,7 )$ . If circle B is translated by $< 3, 1 >$ $<3 ,1 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?