Question

Two circles have the following equations: `(x +6 )^2+(y -5 )^2= 64` and `(x -2 )^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

Circles intersect each other and greatest possible distance between a point on one circle and another point on the other is `29.0416`.

Explanation:

Please refer to details here.

The center of circle `(x+6)^2+(y-5)^2=64` is `(-6,5)` and radius is `8` and center of circle `(x-2)^2+(y+4)^2=81` is `(2,-4)` and radius is `9`.

The distance between centers is `sqrt((2-(-6))^2+(-4-5)^2`

= `sqrt(64+81)=sqrt145=12.0416`

Let the radii of two circles is `r_1` and `r_2` and we also assume that `r_1>r_2` and the distance between centers is `d`.

Then as `r_1+r_2=17 > d=12.0416` and `r_1-r_2=1 < d=12.0416`, the two circles intersect each other and greatest possible distance between a point on one circle and another point on the other is `9+8+12.0416=29.0416`
graph{(x^2+y^2+12x-10y-3)(x^2+y^2-4x+8y-61)=0 [-40, 40, -20, 20]}