Two circles have the following equations (x +5 )^2+(y +6 )^2= 9 and (x +2 )^2+(y -1 )^2= 1 . 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
CW
Sep 18, 2016

One circle does not contain the other. Greatest distance = 11.6158.

Explanation:

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Compare the distance (d) between the centres of the circles to the sum of the radii.

1) If the sum of the radii >d, the circles overlap.
2) If the sum of the radii <d, then no overlap.
3) If d+r_B<= r_A, then Circle A contains Circle B

Given Circle A, centre (-5,-6) and radius r_A=3
Circle B, centre (-2,1), and radius r_B=1

The first step here is to calculate d, use the distance formula :
d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)

where (x_1,y_1) and (x_2,y_2) are 2 coordinate points

here the two points are (-5,-6) and (-2,1) the centres of the circles

let(x_1,y_1)=(-5,-6) and (x_2,y_2)=(-2,1)

d=sqrt(-2-(-5)^2+(1-(-6)^2)
=sqrt(3^2+7^2)=sqrt58=7.6158

Sum of radii = radius of A (r_A)+ radius of B (r_B) = 3+1=4

Since sum of radius <d, then no overlap of the circles
no overlap => no containment

Greatest distance = d(the yellow segment) +r_A+r_B

= 7.6158+3+1=11.6158

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