Two circles have the following equations: $(x +3 )^2+(y -1 )^2= 64$ and $(x -7 )^2+(y +2 )^2= 25$ . 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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Two circles have the following equations: $(x +3 )^2+(y -1 )^2= 64$ and $(x -7 )^2+(y +2 )^2= 25$ . 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 · 2017-11-16T14:39:40

One circle does not contain other. Circles intersect each other. Greatest possible distance between point on one circle and another point on the other is $23.44$

Explanation:

By general:
$(x-x_0)^2+(y-y_0)^2=R^2$

$=> center: (-x_0,-y_0)$
$=> R: sqrt(R^2)$

First let's find the centers and the R of each circle:

For $(x+3)^2+(y-1)^2=64$
Center: $(-3,1)$
$R_1$ : $sqrt64=8$

For $(x-7)^2+(y+2)^2=25$
Center: $(7,-2)$
$R_2$ : $sqrt25=5$

The distance between two points given by Protagoras:
$d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)$
$=>$
The distance between the centers of the circles above given by Protagoras:
$d=sqrt((7-(-3))^2+((-2)-1)^2)=sqrt109~~10.44$

$R_1+R_2=8+5=13$ and $R_1-R_2=8-5=3$

We saw that $R_1+R_2>d$ and $R_1-R_2 < d$ therefore circles intersect each other. Greatest possible distance between point on one circle and another point on the other is $8+5+10.44=23.44$

[For details see here.](https://socratic.org/questions/two-circles-have-the-following-equations-x-4-2-y-3-2-9-and-x-4-2-y-1-2-16-does-o#306687)

graph{((x+3)^2+(y-1)^2-64)((x-7)^2+(y+2)^2-25)=0 [-20, 20, -10, 10]}

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Two circles have the following equations: $(x +3 )^2+(y -1 )^2= 64$ and $(x -7 )^2+(y +2 )^2= 25$ . 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

Explanation:

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Impact of this question

Science

Math

Humanities

... and beyond

Two circles have the following equations: (x +3 )^2+(y -1 )^2= 64 (x +3 )^2+(y -1 )^2= 64 and (x -7 )^2+(y +2 )^2= 25 (x -7 )^2+(y +2 )^2= 25 . 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

Explanation:

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Impact of this question

Two circles have the following equations: $(x +3 )^2+(y -1 )^2= 64$ and $(x -7 )^2+(y +2 )^2= 25$ . 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?