Circle A has a center at #(8 ,-4 )# and a radius of #2 #. Circle B has a center at #(4 ,3 )# and a radius of #4 #. Do the circles overlap? If not what is the smallest distance between them?

Answer 1

The circles do not overlap.
The smallest distance is #=2.1#

The distance between the centre is #d_c#
#sqrt((8-4)^2+(-4-3)^2)#
#=sqrt(16+49)=sqrt65#
The sum of the radii is #r_A+r_B#
#=4+2=6#
As #d_c>(r_A+r_B)#, the circles do not overlap.

The smallest distance between the cicles is

#=sqrt65-6=8.1-6=2.1#

graph{((x-8)^2+(y+4)^2-4)((x-4)^2+(y-3)^2-16)(y-3+7/4(x-4))=0 [-14.24, 14.24, -7.12, 7.12]}

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Answer 2

The circles do not overlap. The distance between their centers is calculated using the distance formula: √((x2 - x1)^2 + (y2 - y1)^2).

For circles A and B, the distance between their centers is √((8 - 4)^2 + (-4 - 3)^2) = √((4)^2 + (-7)^2) = √(16 + 49) = √65 ≈ 8.06 units.

The sum of the radii of circles A and B is 2 + 4 = 6 units.

Since the distance between the centers of the circles is greater than the sum of their radii, the circles do not overlap.

The smallest distance between them is the difference between the distance between their centers and the sum of their radii: 8.06 - 6 = 2.06 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.

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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