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?

1 Answer
Shwetank Mauria
Sep 7, 2016

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.](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#306687)

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]}

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