Two circles have the following equations $(x +5 )^2+(y -2 )^2= 36$ and $(x +2 )^2+(y -1 )^2= 81$ . 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?

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Answer 1 · 2016-12-26T10:59:01

The circles overlap
The greatest possible distance is $=18.16$

We need the equation of a circle, center $(a,b)$ and radius $r$ is

$(x-a)^2+(y-b)^2=r^2$

The distance between 2 points, $(x_1,y_1)$ and $(x_2,y_2)$ is

$=sqrt((x_2-x_1)^2+(x_2-y_2)^2)$

We need to find the distance between the centres of the circles and compare this to the sum of the radii.

The centers are $C_A=(-5,2)$ and $C_B=(-2,1)$

The distance between the centers is

$d=sqrt((-5--2)^2+(2-1)^2)$

$=sqrt(9+1)=sqrt10=3.16$

The sum of the radii is $=6+9=15$

Therefore,

$d<$ sum of radii

so,

The circles overlap

The greatest possible distance is $=9+6+3.16=18.16$

graph{((x+5)^2+(y-2)^2-36)((x+2)^2+(y-1)^2-81)(y-2+1/3(x+5)) = 0 [-19.22, 9.27, -5.39, 8.86]}

Two circles have the following equations (x +5 )^2+(y -2 )^2= 36 (x +5 )^2+(y -2 )^2= 36 and (x +2 )^2+(y -1 )^2= 81 (x +2 )^2+(y -1 )^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?