Circle A has a radius of $1$ $1$ and a center at $(3, 3)$ $(3 ,3 )$ . Circle B has a radius of $3$ $3$ and a center at $(1, 7)$ $(1 ,7 )$ . 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 $1$ $1$ and a center at $(3, 3)$ $(3 ,3 )$ . Circle B has a radius of $3$ $3$ and a center at $(1, 7)$ $(1 ,7 )$ . 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 · 2017-11-05T20:21:40

$no overlap$ $"no overlap"$

Explanation:

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

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

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

$before calculating d we require to find the coordinates$ $"before calculating d we require to find the coordinates"$
$of the centre of circle B under the given translation$ $"of the centre of circle B under the given translation"$
$which does not change the shape of the circle only$ $"which does not change the shape of the circle only"$
$it's position$ $"it's position"$

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

$(1, 7) \to (1 + 2, 7 + 4) \to (3, 11) \leftarrow new centre of B$ $(1,7)to(1+2,7+4)to(3,11)larrcolor(blue)"new centre of B"$

$note that the coordinates of the 2 centres have the same$ $"note that the coordinates of the 2 centres have the same"$
$x-coordinate and so lie on the vertical line x = 3$ $"x-coordinate and so lie on the vertical line "x=3$

$Hence d is the difference in the y-coordinates$ $"Hence d is the difference in the y-coordinates"$

$\Rightarrow d = 11 - 3 = 8$ $rArrd=11-3=8$

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

$since sum of radii < d then no overlap of circles$ $"since sum of radii"< d" then no overlap of circles"$
graph{((x-3)^2+(y-3)^2-1)((x-3)^2+(y-11)^2-9)=0 [-40, 40, -20, 20]}

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Circle A has a radius of $1$ $1$ and a center at $(3, 3)$ $(3 ,3 )$ . Circle B has a radius of $3$ $3$ and a center at $(1, 7)$ $(1 ,7 )$ . 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 11 and a center at (3,3)(3 ,3 ). Circle B has a radius of 33 and a center at (1,7)(1 ,7 ). 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?

1 Answer

Explanation:

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Circle A has a radius of $1$ $1$ and a center at $(3, 3)$ $(3 ,3 )$ . Circle B has a radius of $3$ $3$ and a center at $(1, 7)$ $(1 ,7 )$ . 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?