Question

Two circles have the following equations: `(x +3 )^2+(y -1 )^2= 64` and `(x -7 )^2+(y +2 )^2= 25`. 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

One circle does not contain other. Circles intersect each other. Greatest possible distance between point on one circle and another point on the other is `23.44`

Explanation:

By general:
`(x-x_0)^2+(y-y_0)^2=R^2`

`=> center: (-x_0,-y_0)`
`=> R: sqrt(R^2)`

First let's find the centers and the R of each circle:

For `(x+3)^2+(y-1)^2=64`
Center: `(-3,1)`
`R_1`: `sqrt64=8`

For `(x-7)^2+(y+2)^2=25`
Center: `(7,-2)`
`R_2`: `sqrt25=5`

The distance between two points given by Protagoras:
`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`
`=>`
The distance between the centers of the circles above given by Protagoras:
`d=sqrt((7-(-3))^2+((-2)-1)^2)=sqrt109~~10.44`

`R_1+R_2=8+5=13` and `R_1-R_2=8-5=3`

We saw that `R_1+R_2>d` and `R_1-R_2 < d` therefore circles intersect each other. Greatest possible distance between point on one circle and another point on the other is `8+5+10.44=23.44`

For details see here.

graph{((x+3)^2+(y-1)^2-64)((x-7)^2+(y+2)^2-25)=0 [-20, 20, -10, 10]}