# Two circles have the following equations #(x +5 )^2+(y -2 )^2= 36 # and #(x +2 )^2+(y -1 )^2= 81 #. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?

The circles overlap

The greatest possible distance is

We need to find the distance between the centres of the circles and compare this to the sum of the radii.

The distance between the centers is

Therefore,

so,

The circles overlap

graph{((x+5)^2+(y-2)^2-36)((x+2)^2+(y-1)^2-81)(y-2+1/3(x+5)) = 0 [-19.22, 9.27, -5.39, 8.86]}

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No, one circle does not contain the other. The greatest possible distance between a point on one circle and another point on the other can be found by calculating the sum of their radii and then subtracting the distance between their centers. In this case, the radii of the circles are 6 and 9 respectively, and the distance between their centers is the square root of ((2-(-5))^2 + (1-2)^2 = \sqrt{49+1} = \sqrt{50}). Therefore, the greatest possible distance between a point on one circle and another point on the other is (6 + 9 - \sqrt{50}).

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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.

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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