Two circles have the following equations: (x +6 )^2+(y -5 )^2= 64 and (x -9 )^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
Rui D.
Apr 6, 2016

The circles overlap but the small circle isn't contained in the big one. The biggest possible distance between 2 points in the 2 circles is sqrt226+17~=32.033

Explanation:

(x+6)^2+(y-5)^2=64 => r_1=8, C_1 (-6,5)
(x-9)^2+(y-4)^2=81 => r_2=9, C_2 (9,4)

r_1+r_2=17
r_2-r_1=1

d_(C_1C_2)=sqrt((9+6)^2+(4-5)^2)=sqrt(225+1)=sqrt(226)~=15.033

Since r_2-r_1 < d_(C_1C_2) < r_1+r_2
the circles overlap but circle 1 isn't contained in circle 2.

The greatest possible distance between 2 points in the 2 circles is
=d_(C_1C_2)+r_1+r_2=sqrt226+8+9=sqrt226+17~=32.033

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