# Circle A has a center at (2 ,1 ) and an area of 15 pi. Circle B has a center at (8 ,12 ) and an area of 9 pi. Do the circles overlap? If not, what is the shortest distance between them?

No, the circles do not overlap
shortest distance between them$= \sqrt{157} - \sqrt{15} - 3 = 5.65698$

#### Explanation:

Distance between centers:
$d = \sqrt{{\left({x}_{a} - {x}_{b}\right)}^{2} + {\left({y}_{a} - {y}_{b}\right)}^{2}}$
$d = \sqrt{{\left(2 - 8\right)}^{2} + {\left(1 - 12\right)}^{2}}$
$d = \sqrt{{\left(- 6\right)}^{2} + {\left(- 11\right)}^{2}}$
$d = \sqrt{36 + 121}$
$d = \sqrt{157}$

Shortest distance between the two circles $= d - \left({r}_{a} + {r}_{b}\right)$

$= \sqrt{157} - \left(\sqrt{15} + \sqrt{9}\right) = 5.65698$

Let us see the graph of the circles ${\left(x - 2\right)}^{2} + {\left(y - 1\right)}^{2} = 15$ and
${\left(x - 8\right)}^{2} + {\left(y - 12\right)}^{2} = 9$
graph{((x-2)^2+(y-1)^2-15)((x-8)^2+(y-12)^2-9)=0[-20,30,-10,16]}

God bless ....I hope the explanation is useful..