Two circles have the following equations: (x +6 )^2+(y -1 )^2= 49 (x+6)2+(y1)2=49 and (x -9 )^2+(y -4 )^2= 81 (x9)2+(y4)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
Feb 18, 2017

The circles overlap and the greatest distance is =31.3=31.3

Explanation:

We compare the sum of the radii to the distance between the centers,

Radius of first circle r_1=7r1=7

Radius of second circle r_2=9r2=9

r_1+r_2=9+7=16r1+r2=9+7=16

The center of the first circle is O=(-6,1)O=(6,1)

The center of the second circle is O'=(9,4)

The distance between the centers is

d_(OO') = sqrt((9--6)^2+(4-1)^2)

=sqrt(225+9)

=sqrt234=15.3

Therefore,

d_(OO') < r_1+r_2

So,

the circles overlap

The greatest distance is =15.3+7+9=31.3

graph{((x+2)^2+(y-1)^2-49)((x-9)^2+(y-4)^2-81)(y-4-1/5(x-9))=0 [-22.8, 22.83, -11.4, 11.4]}