Question

Two circles have the following equations: `(x -1 )^2+(y -2 )^2= 9` and `(x +6 )^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

The two circles are outside each other i.e. one circle is not contained in the other and greatest distance between a point on one circle and another point on the other is `16.06`

Explanation:

The center of `(x-1)^2+(y-2)^2=9` is`(1,2)` and radius is `3`.

and center of `(x+6)^2+(y+2)^2=25` is`(-6,-2)` and radius is `5`.

The distance between centers is `sqrt((1-(-6))^2+(2-(-2))^2)`

= `sqrt(7^2+4^2)=sqrt(49+16)=sqrt65=8.06`

Hence sum of raadii is `8` and is less than distance between their centers.

Hence the two circles are outside each other i.e. one circle is not contained in the other.

and greatest distance between a point on one circle and another point on the other is`3+5+8.06=16.06`

For details see https://socratic.org/questions/two-circles-have-the-following-equations-x-4-2-y-3-2-9-and-x-4-2-y-1-2-16-does-o

graph{((x-1)^2+(y-2)^2-9)((x+6)^2+(y+2)^2-25)=0 [-2.853, -0.353, -0.085, 1.165]}

graph{((x-1)^2+(y-2)^2-9)((x+6)^2+(y+2)^2-25)=0 [-12.29, 7.71, -5.34, 4.66]}