Circle A has a radius of $2$ $2$ and a center of $(6, 5)$ $(6 ,5 )$ . Circle B has a radius of $1$ $1$ and a center of $(3, 4)$ $(3 ,4 )$ . If circle B is translated by $< 1, 3 >$ $<1 ,3 >$ , 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 $(6, 5)$ $(6 ,5 )$ . Circle B has a radius of $1$ $1$ and a center of $(3, 4)$ $(3 ,4 )$ . If circle B is translated by $< 1, 3 >$ $<1 ,3 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Answer 1 · 2018-07-22T14:15:44

$circles overlap$ $"circles overlap"$

Explanation:

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

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

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

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

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

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

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

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

$let (x_{1}, y_{1}) = (6, 5) and (x_{2}, y_{2}) = (4, 7)$ $"let "(x_1,y_1)=(6,5)" and "(x_2,y_2)=(4,7)$

$d = \sqrt{{(4 - 6)}^{2} + {(7 - 5)}^{2}} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83$ $d=sqrt((4-6)^2+(7-5)^2)=sqrt(4+4)=sqrt8~~2.83$

$sum of radii = 2 + 1 = 3$ $"sum of radii "=2+1=3$

$since sum of radii > d then circles overlap$ $"Since sum of radii">d" then circles overlap"$
graph{((x-6)^2+(y-5)^2-4)((x-4)^2+(y-7)^2-1)=0 [-40, 40, -20, 20]}

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Circle A has a radius of $2$ $2$ and a center of $(6, 5)$ $(6 ,5 )$ . Circle B has a radius of $1$ $1$ and a center of $(3, 4)$ $(3 ,4 )$ . If circle B is translated by $< 1, 3 >$ $<1 ,3 >$ , 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 (6,5)(6 ,5 ). Circle B has a radius of 11 and a center of (3,4)(3 ,4 ). If circle B is translated by <1,3><1 ,3 >, 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 $(6, 5)$ $(6 ,5 )$ . Circle B has a radius of $1$ $1$ and a center of $(3, 4)$ $(3 ,4 )$ . If circle B is translated by $< 1, 3 >$ $<1 ,3 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?