# Two circles have the following equations #(x -1 )^2+(y -4 )^2= 64 # and #(x +3 )^2+(y +1 )^2= 9 #. 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?

See below.

It is possible to ascertain whether two circles do not touch, only touch at one point, or intersect twice.

The circles do not touch if their total radii are less than or equal to the distance between their centers; if the total radii are greater than or equal to the distance between their centers, the circles intersect at two points; and if the total radii are equal to the distance between their centers, the circles touch at one point.

Using the distance formula, we can first determine the distance between their centers.

The formula for distance is:

Total radii:

The total distance between the centers of the two circles and their radii will determine the maximum distance between a point on one circle and a point on the other:

3 .d.p.

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The given circles do not contain each other. The greatest possible distance between a point on one circle and another point on the other can be found by determining the distance between their centers and then subtracting the sum of their radii. The distance between the centers is calculated using the distance formula, and the sum of the radii is added to this distance to find the greatest possible distance between points on the circles.

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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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- Why is the area of a circle #pi r^2#?

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