# Two circles have the following equations (x -2 )^2+(y -4 )^2= 36  and (x +8 )^2+(y +3 )^2= 49 . Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?

Jul 12, 2017

The circles overlap and the greatest distance is $= 25.21$

#### Explanation:

The center of circle $A$ is ${C}_{A} = \left(2 , 4\right)$ and radius ${r}_{A} = 6$

The center of circle $B$ is ${C}_{B} = \left(- 8 , - 3\right)$ and radius ${r}_{B} = 7$

The distance between the centers is

${C}_{A} {C}_{B} = \sqrt{{\left(10\right)}^{2} + {\left(7\right)}^{2}} = \sqrt{149} = 12.21$

The sum of the radii is

$R = {r}_{A} + {r}_{B} = 6 + 7 = 13$

As,

$R > {C}_{A} {C}_{B}$, the circles overlap

The greatest distance is $= 12.21 + 6 + 7 = 25.21$

graph{((x-2)^2+(y-4)^2-36)((x+8)^2+(y+3)^2-49)(y-4-7/10(x-2))=0 [-32.49, 32.46, -16.24, 16.25]}