Question

Two circles have the following equations: `(x +6 )^2+(y -5 )^2= 64` 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 but the small circle isn't contained in the big one. The biggest possible distance between 2 points in the 2 circles is `sqrt226+17~=32.033`

Explanation:

`(x+6)^2+(y-5)^2=64` => `r_1=8`, `C_1 (-6,5)`
`(x-9)^2+(y-4)^2=81` => `r_2=9`, `C_2 (9,4)`

`r_1+r_2=17`
`r_2-r_1=1`

`d_(C_1C_2)=sqrt((9+6)^2+(4-5)^2)=sqrt(225+1)=sqrt(226)~=15.033`

Since `r_2-r_1 < d_(C_1C_2) < r_1+r_2`
the circles overlap but circle 1 isn't contained in circle 2.

The greatest possible distance between 2 points in the 2 circles is
`=d_(C_1C_2)+r_1+r_2=sqrt226+8+9=sqrt226+17~=32.033`