Circle A has a center at #(3 ,7 )# and a radius of #4 #. Circle B has a center at #(4 ,-2 )# and a radius of #6 #. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

circles overlap.

What we have to do here is compare the distance ( d ) 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 no overlap

To calculate d, use the #color(blue)"distance formula"#
#color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))# where # (x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points."#

The 2 points here are (3 ,7) and (4 ,-2) the centres of the circles.

let #(x_1,y_1)=(3,7)" and " (x_2,y_2)=(4,-2)#
#d=sqrt((4-3)^2+(-2-7)^2)=sqrt(1+81)=sqrt82≈9.055#

sum of radii = radius of A + radius of B = 4 + 6 = 10

Since sum of radii > d , then circles overlap graph{(y^2-14y+x^2-6x+42)(y^2+4y+x^2-8x-16)=0 [-40, 40, -20, 20]}

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Answer 2

Yes, the circles overlap. The smallest distance between the circles occurs at the points where the line connecting the centers of the circles intersects the circles. This distance can be calculated using the distance formula. The distance between the centers of the circles is the square root of the sum of the squares of the differences in their x-coordinates and y-coordinates. Then, subtracting the sum of the radii from this distance gives the smallest distance between the circles. Applying this formula, the distance between the centers of circles A and B is ( \sqrt{(4-3)^2 + (-2-7)^2} = \sqrt{1^2 + (-9)^2} = \sqrt{1 + 81} = \sqrt{82} \approx 9.06 ). Subtracting the sum of the radii (4 + 6 = 10) from this distance, we get ( \sqrt{82} - 10 \approx -0.94 ). Since this result is negative, it indicates that the circles overlap.

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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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