Question

Two circles have the following equations: `(x +6 )^2+(y -1 )^2= 49` and `(x -9 )^2+(y -4 )^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 and the greatest distance is `=31.3`

Explanation:

We compare the sum of the radii to the distance between the centers,

Radius of first circle `r_1=7`

Radius of second circle `r_2=9`

`r_1+r_2=9+7=16`

The center of the first circle is `O=(-6,1)`

The center of the second circle is `O'=(9,4)`

The distance between the centers is

`d_(OO') = sqrt((9--6)^2+(4-1)^2)`

`=sqrt(225+9)`

`=sqrt234=15.3`

Therefore,

`d_(OO') < r_1+r_2`

So,

the circles overlap

The greatest distance is `=15.3+7+9=31.3`

graph{((x+2)^2+(y-1)^2-49)((x-9)^2+(y-4)^2-81)(y-4-1/5(x-9))=0 [-22.8, 22.83, -11.4, 11.4]}