# Circle A has a center at (4 ,-8 ) and a radius of 3 . Circle B has a center at (-2 ,-2 ) and a radius of 2 . Do the circles overlap? If not, what is the smallest distance between them?

Jul 6, 2017

The circles do not overlap and the shortest distance is $= 3.5$

#### Explanation:

The distance between the centers is

${O}_{A} {O}_{B} = \sqrt{{\left(- 2 - \left(4\right)\right)}^{2} + {\left(- 2 - \left(- 8\right)\right)}^{2}}$

$= \sqrt{36 + 36}$

$= \sqrt{72} = 8.5$

The sum of the radii is

${r}_{A} + {r}_{B} = 3 + 2 = 5$

As,

${O}_{A} {O}_{B} > \left({r}_{A} + {r}_{B}\right)$

The circles do not overlap.

The smallest distance is

$d = 8.5 - 5 = 3.5$

graph{((x-4)^2+(y+8)^2-9)((x+2)^2+(y+2)^2-4)(y+x+4)=0 [-25.84, 25.46, -16.57, 9.1]}

Jul 6, 2017

$\text{no overlap } d \approx 3.485$

#### Explanation:

What we have to do here is $\textcolor{b l u e}{\text{compare }}$the distance ( d) between the centres of the circles with the $\textcolor{b l u e}{\text{ sum of the radii}}$

• " if sum of radii " > d" then circles overlap"

• " if sum of radii "< d" then no overlap"

$\text{to calculate d use the "color(blue)"distance formula}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
$\left({x}_{1} , {y}_{1}\right) , \left({x}_{2} , {y}_{2}\right) \text{ are 2 coordinate points}$

$\text{the points are } \left({x}_{1} , {y}_{1}\right) = \left(4 , - 8\right) , \left({x}_{2} , {y}_{2}\right) = \left(- 2 , - 2\right)$

$d = \sqrt{{\left(- 2 - 4\right)}^{2} + {\left(- 2 + 8\right)}^{2}} = \sqrt{72} \approx 8.485$

$\text{sum of radii } = 3 + 2 = 5$

$\text{since sum of radii "< d" then no overlap}$

$\text{smallest distance "=d-" sum of radii}$

$= 8.485 - 5 = 3.485$
graph{(y^2+16y+x^2-8x+71)(y^2+4y+x^2+4x+4)=0 [-25.31, 25.32, -12.66, 12.65]}