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

Oct 25, 2017

$\therefore$Circles A and B do not intersect at all.

#### Explanation:

Two circles overlap if they intersect at two points .
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Two circle $A \left(O 1 , r 1\right) \text{ "and " } B \left(O 2 , r 2\right)$ intersect at two points
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$\iff d \left(O 1 , O 2\right) < r 1 + r 2 \text{ }$ where $\text{ } d \left(O 1 , O 2\right)$ is the distance
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between the two centres.
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$d \left(O 1 , O 2\right) = \sqrt{{\left({y}_{O 2} - {y}_{O 1}\right)}^{2} + {\left({x}_{O 2} - {X}_{O 1}\right)}^{2}}$
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$d \left(O 1 , O 2\right) = \sqrt{{\left(- 2 - 3\right)}^{2} + {\left(- 1 - 2\right)}^{2}}$
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$d \left(O 1 , O 2\right) = \sqrt{{\left(- 5\right)}^{2} + {\left(- 3\right)}^{2}}$
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$d \left(O 1 , O 2\right) = \sqrt{25 + 9} = \sqrt{34}$
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$r 1 + r 2 = 1 + 4 = 5$
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$d \left(O 1 , O 2\right) > r 1 + r 2$
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$\therefore$Circles A and B do not intersect at all.