Two circles have the following equations (x +5 )^2+(y -2 )^2= 36 and (x -1 )^2+(y +5 )^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
Douglas K.
Oct 2, 2016

The larger circle does not contain the entire smaller circle. The greatest possible distance between two points is on the line between the centers: 15 + sqrt85

Explanation:

Rewrite the equations is standard form:
(x - -5)² + (y - 2)² = 6²
(x - 1)² + (y - -5)² = 9²

The centers are at (-5, 2) and (1, -5) respectively.

Let's check the distance between the centers:

d = sqrt((-5 - 1)² + (2 - -5)²)

d = sqrt(36 + 49)

d = sqrt(85)

Because they are only just a bit farther apart than the radius of the larger circle, they surely intersect at 2 points.

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