# Circle A has a center at (1 ,8 ) and an area of 18 pi. Circle B has a center at (8 ,1 ) and an area of 45 pi. Do the circles overlap?

Nov 21, 2016

The circles overlap

#### Explanation:

Circle A
$a r e a = \pi {r}_{A}^{2} = 18 \pi$

So, ${r}_{A} = \sqrt{18} = 3 \sqrt{2}$

Circle B
$a r e a = \pi {r}_{B}^{2} = 45 \pi$

So, ${r}_{B} = \sqrt{45} = 3 \sqrt{5}$

The distance between the centers of the circles $A \left({x}_{A} , {y}_{A}\right)$ and $B \left({x}_{B} , {y}_{B}\right)$

$d = \sqrt{{\left({x}_{B} - {x}_{A}\right)}^{2} + {\left({y}_{b} - {y}_{A}\right)}^{2}}$

$d = \sqrt{{\left(8 - 1\right)}^{2} + {\left(1 - 8\right)}^{2}} = \sqrt{{7}^{2} + {7}^{2}} = 7 \sqrt{2} = 9.9$

The sum of the radii $= {r}_{A} + {r}_{B} = 3 \sqrt{2} + 3 \sqrt{5} = 10.95$

Therefore, $d \le {r}_{A} + {r}_{B}$
So, the circles overlap

The equations of the circles are

${\left(x - 1\right)}^{2} + {\left(y - 8\right)}^{2} = 18$

${\left(x - 8\right)}^{2} + {\left(y - 1\right)}^{2} = 45$
graph{((x-1)^2+(y-8)^2-18)((x-8)^2+(y-1)^2-45)=0 [-22.45, 23.15, -5.73, 17.09]}