# Circle A has a center at (1 ,4 ) and a radius of 2 . Circle B has a center at (9 ,3 ) and a radius of 1 . Do the circles overlap? If not what is the smallest distance between them?

Aug 7, 2016

$\textcolor{g r e e n}{\text{They do not overlap.}}$
color(green)("The smallest distance between them is: "sqrt(65)-3"~~5.06

#### Explanation:

If the distance between centres is less than the sum of the radii then they overlap.

Let point 1 be ${P}_{1} \to \left({x}_{1} , {y}_{1}\right) = \left(1 , 4\right)$
Let point 2 be ${P}_{2} \to \left({x}_{2} , {y}_{2}\right) = \left(9 , 3\right)$
Let the distance between centres be $d$

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$\textcolor{b l u e}{\text{Determine distance between centres}}$

${x}_{\text{difference}} = {x}_{2} - {x}_{1} = 9 - 1 = 8$
${y}_{\text{difference}} = {y}_{2} = {y}_{1} = 3 - 4 = - 1$

Using Pythagoras $d = \sqrt{{8}^{2} + {\left(- 1\right)}^{2}} = \sqrt{65}$

This is approximately $8$
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$\textcolor{b l u e}{\text{Determine if they over lap}}$

The sum of the radii is $2 + 1 = 3$

So if the sum of the radii is 3 and the distance between centres is very close to 8 $\textcolor{g r e e n}{\underline{\text{ they do not overlap.}}}$

color(green)("The smallest distance between them is: "sqrt(65)-3"