Two circles have the following equations $(x +2 )^2+(y -5 )^2= 16$ and $(x +4 )^2+(y -3 )^2= 49$ . 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 $(x +2 )^2+(y -5 )^2= 16$ and $(x +4 )^2+(y -3 )^2= 49$ . 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 · 2016-09-06T09:11:30

Smaller circle (radius $4$ ) is contained in larger circle (radius $7$ ). The greatest possible distance between a point on one circle and another point on the other is $13.8284$ .

Explanation:

The center of circle $(x+2)^2+(y-5)^2=16$ is $(-2,5)$ and radius is $4$ and center of circle $(x+4)^2+(y-3)^2=49$ is $(-4,3)$ and radius is $7$ .

The distance between centers is $sqrt(((-4)-(-2))^2+(3-5)^2$

= $sqrt(4+4)=sqrt8=2sqrt2=2.8284$

If the radii of two circles is $r_1$ and $r_2$ and we also assume that $r_1>r_2$ and the distance between centers is $d$ , then

A - if $r_1+r_2=d$ , they touch each other externally and greatest possible distance between a point on one circle and another point on the other is $2(r_1+r_2)$ and smallest possible distance between a point on one circle and another point on the other is $0$ .

B - if $r_1+r_2 < d$ , they do not touch each other and are outside each other (i.e. one is not contained in other) and greatest possible distance between a point on one circle and another point on the other is $r_1+r_2+d$ and smallest possible distance between a point on one circle and another point on the other is $d-r_1-r_2$ .

C - if $r_1+r_2 > d$ and $r_1-r_2=d$ , they touch each other internally and smaller circle is contained in other and greatest possible distance between a point on one circle and another point on the other is $2r_2$ and smallest distance is $0$ .

D - if $r_1+r_2 > d$ and $r_1-r_2>d$ , smaller circle lies inside the larger circle and greatest possible distance between a point on one circle and another point on the other is $r_1+r_2+d$ and smallest distance is $r_1-r_2-d$ .

E - if $r_1+r_2 > d$ and $r_1-r_2 < d$ , the two circles intersect each other and greatest possible distance between a point on one circle and another point on the other is $r_1+r_2+d$ and smallest distance is $0$ .

Now, here as $r_1+r_2=11>d=2.8284$ and $r_1-r_2=3>d=2.8284$ . Hence, smaller circle lies inside the larger circle and hence is contained in bigger circle. The greatest possible distance between a point on one circle and another point on the other is $r_1+r_2+d=7+4+2.8284=13.8284$ .

graph{(x^2+y^2+4x-10y+13)(x^2+y^2+8x-6y-24)=0 [-22, 18, -5.36, 14.64]}

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Two circles have the following equations $(x +2 )^2+(y -5 )^2= 16$ and $(x +4 )^2+(y -3 )^2= 49$ . 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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Science

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Humanities

... and beyond

Two circles have the following equations (x +2 )^2+(y -5 )^2= 16 (x +2 )^2+(y -5 )^2= 16 and (x +4 )^2+(y -3 )^2= 49 (x +4 )^2+(y -3 )^2= 49 . 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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Two circles have the following equations $(x +2 )^2+(y -5 )^2= 16$ and $(x +4 )^2+(y -3 )^2= 49$ . 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?