# Circle A has a center at #(-6 ,4 )# and a radius of #9 #. Circle B has a center at #(1 ,-5 )# and a radius of #5 #. Do the circles overlap? If not, what is the smallest distance between them?

Yes, they overlap.

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The circles do overlap. The distance between the centers of the circles is ( \sqrt{(1 - (-6))^2 + (-5 - 4)^2} = \sqrt{7^2 + (-9)^2} = \sqrt{49 + 81} = \sqrt{130} \approx 11.4 ). Since this distance is less than the sum of the radii of the circles (9 + 5 = 14), the circles overlap.

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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.

- Circle A has a center at #(5 ,4 )# and an area of #16 pi#. Circle B has a center at #(12 ,8 )# and an area of #9 pi#. Do the circles overlap? If not, what is the shortest distance between them?
- A triangle has corners at #(4 ,6 )#, #(3 ,9 )#, and #(6 ,5 )#. How far is the triangle's centroid from the origin?
- Given #C_1->y^2+x^2-4x-6y+9=0#, #C_2->y^2+x^2+10x-16y+85=0# and #L_1->x+2y+15=0#, determine #C->(x-x_0)^2+(y-y_0)^2-r^2=0# tangent to #C_1,C_2# and #L_1#?
- A line passes through #(9 ,2 )# and #( 4, 8 )#. A second line passes through #( 4, 1 )#. What is one other point that the second line may pass through if it is parallel to the first line?
- A line passes through #(3 ,5 )# and #(6 ,8 )#. A second line passes through #(7 ,4 )#. What is one other point that the second line may pass through if it is parallel to the first line?

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