Question

Two circles have the following equations `(x -1 )^2+(y -4 )^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?

Answer 1

Because the distance between the centers is not less than the difference between the two radii, we must conclude that the larger circle partially contains the smaller circle.

Explanation:

Put is standard form:

`(x - 1)^2 + (y - 4)^2 = 6^2`
`(x - 1)^2 + (y - -5)^2 = 9^2`

Please notice that the x coordinate for center of both circles is 1, therefore, the distance between the centers is the difference between their y coordinates.

`(4 - -5) = 9`

Because the distance between the center is not less than the difference between the two radii, we must conclude that the larger circle partially contains the smaller circle.