Circle A has a radius of #5 # and a center of #(8 ,2 )#. Circle B has a radius of #3 # and a center of #(3 ,7 )#. If circle B is translated by #<4 , 8 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

#"no overlap "~~5.01#

#"What we have to do here is compare the distance d "# #"between the centres to the sum of the radii"#
#• " if sum of radii">d" then circles overlap"#
#• " if sum of radii"< d" then no overlap"#
#"Before calculating d we require to find the new centre"# #"of B under the given translation"#
#"under the translation "< 4,8>#
#(3,7)to(3+4,7+8)to(7,15)larrcolor(red)"new centre of B"#
#"to calculate d use the "color(blue)"distance formula"#
#•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#
#"let "(x_1,y_1)=(8,2)" and "(x_2,y_2)=(7,15)#
#d=sqrt((7-8)^2+(15-2)^2)=sqrt(1+169)=sqrt170~~13.01#
#"sum of radii "=5+3=8#
#"Since sum of radii"< d" then no overlap"#
#"min distance "=d-" sum of radii"#
#color(white)(xxxxxxxxxx)=13.01-8=5.01# graph{((x-8)^2+(y-2)^2-25)((x-7)^2+(y-15)^2-9)=0 [-40, 40, -20, 20]}
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Answer 2

To determine if circle B overlaps circle A after being translated, we need to find the distance between the centers of the circles after the translation and compare it to the sum of the radii of the circles.

The center of circle B after translation is obtained by adding the translation vector to the original center coordinates of circle B:

New center of circle B = (3 + 4, 7 + 8) = (7, 15)

Now, we calculate the distance between the centers of the circles:

Distance between centers = √[(x2 - x1)^2 + (y2 - y1)^2] = √[(7 - 8)^2 + (15 - 2)^2] = √[(-1)^2 + (13)^2] = √(1 + 169) = √170

Now, we compare this distance to the sum of the radii of the circles:

Sum of radii = 5 + 3 = 8

Since the distance between the centers (sqrt(170)) is greater than the sum of the radii (8), the circles do not overlap.

To find the minimum distance between points on both circles, we subtract the radii of the circles from the distance between their centers:

Minimum distance = Distance between centers - Sum of radii = √170 - 8 = √170 - 8 ≈ 3.96 units

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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