# Two circles have the following equations (x -8 )^2+(y -2 )^2= 36  and (x -1 )^2+(y +5 )^2= 81 . 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?

Oct 26, 2016

No. $15 + 7 \sqrt{2}$

#### Explanation:

C_1=(8;2); ${r}_{1} = 6$
C_2=(1;-5); ${r}_{2} = 9$
$\overline{{C}_{1} {C}_{2}} = \sqrt{{\left(8 - 1\right)}^{2} + {\left(2 + 5\right)}^{2}} = 7 \sqrt{2} > 9$
So ${C}_{1}$ is outside circle 2.

You obtain the maximum possible distance adding
$\overline{{C}_{1} {C}_{2}} + {r}_{1} + {r}_{2} = 15 + 7 \sqrt{2}$