Question

Two circles have the following equations `(x -2 )^2+(y -4 )^2= 36` and `(x +8 )^2+(y -7 )^2= 49`. 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 the other rather they intersect each other. The greatest possible distance between a point on one circle and another point on the other is `23.44`.

Explanation:

`(x-2)^2+(y-4)^2=36` is a circle with center at `(2,4)` and radius `6`.

`(x+8)^2+(y-7)^2=49` is a circle with center at `(-8,7)` and radius `7`

The distance between centers of two circles is `sqrt((2-(-8))^2+(4-7)^2)` or `sqrt(10^2+3^2)=sqrt109=10.44`

As the distance between centers is `10.44` and radius of two circles is `6` and `7` but distance is less than sum of radii, one circle does not contain the other rather they intersect each other.

The greatest possible distance between a point on one circle and another point on the other is `10.44+7+6=23.44`.

graph{((x-2)^2+(y-4)^2-36)((x+8)^2+(y-7)^2-49)=0 [-30, 30, -15, 15]}