Question

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?

Answer 1

The circles overlap.

Explanation:

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]}