# Circle A has a center at #(3 ,2 )# and a radius of #1 #. Circle B has a center at #(1 ,3 )# and a radius of #4 #. Do the circles overlap? If not, what is the smallest distance between them?

circle A enclosed inside circle B

What we have to do here is compare the distance (d) between the centres with the sum and difference of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

• If difference of radii > d , then 1 circle inside other

The 2 points here are the centres (3 ,2) and (1 ,3)

sum of radii = radius of A + radius of B = 1 + 4 = 5

difference of radii = radius of B - radius of A = 4 - 1 = 3

Since difference of radii > d , then circle A is enclosed inside circle B. graph{(y^2-4y+x^2-6x+12)(y^2-6y+x^2-2x-6)=0 [-10, 10, -5, 5]}

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The circles do not overlap. The smallest distance between them is 2 units.

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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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- Circle A has a center at #(12 ,9 )# and an area of #25 pi#. Circle B has a center at #(3 ,1 )# and an area of #67 pi#. Do the circles overlap?
- Circle A has a center at #(1 ,5 )# and an area of #24 pi#. Circle B has a center at #(8 ,4 )# and an area of #66 pi#. Do the circles overlap?
- A line passes through #(2 ,4 )# and #(7 ,5 )#. A second line passes through #(3 ,5 )#. What is one other point that the second line may pass through if it is parallel to the first line?
- A line passes through #(8 ,5 )# and #(2 ,1 )#. A second line passes through #(3 ,5 )#. What is one other point that the second line may pass through if it is parallel to the first line?

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