Two circles have the following equations $(x -1 )^2+(y -4 )^2= 36$ and $(x +5 )^2+(y +5 )^2= 81$ . 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?

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Answer 1 · 2017-06-13T13:09:34

The circles overlap.

The radius of circle $A$ is

$r_A=sqrt(36)=6$

The radius of circle $B$ is

$r_B=sqrt(81)=9$

The distance between the center $=(1,4)$ of circle $A$ and the center $(-5,-5)$ of circle $B$ is

$d=sqrt((1+5)^2+(4+5)^2)$

$=sqrt(6^2+9^2)$

$sqrt(36+81)=sqrt117$

$=10.81$

The sum of the radii is

$r_A+r_B=6+9=15$

Therefore,

As $(r_A+r_B )> d$

the circles overlap.
graph{((x-1)^2+(y-4)^2-36)((x+5)^2+(y+5)^2-81)=0 [-28.87, 28.88, -14.43, 14.43]}

Two circles have the following equations (x -1 )^2+(y -4 )^2= 36 (x -1 )^2+(y -4 )^2= 36 and (x +5 )^2+(y +5 )^2= 81 (x +5 )^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?