Two circles have the following equations $(x -1 )^2+(y -4 )^2= 64$ and $(x +3 )^2+(y -4 )^2= 9$ . 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 · 2016-09-26T03:10:33

Yes, one circle contains the other.

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Circle A, $(x-1)^2+(y-4)^2=64$ , center $(1,4)$ , radius $r_A=8$
Circle B, $(x+3)^2+(y-4)^2=9$ , center $(-3,4)$ , radius $r_B=3$

1) calculate d the distance between the centres of the circles, use the distance formula :
$d=sqrt((x2−x1)^2+(y2−y1))^2$

where $(x1,y1)and(x2,y2)$ are $(1,4)$ , and $(-3,4)$

$d=sqrt((−3−1)^2+(4-4)^2)=sqrt(16)=4$

2) calculate the sum of the radii $(r_A+r_B)$

Sum of radii = $r_ A + r_B =8+3=11$

3) calculate the difference of the radii $(r_A-r_B)$

Difference of Radii $r_ A- r_B = 8-3=5$

3 )compare the distance d between the centres of the circles to the sum of the radii and to the difference of the radii.

1) If $r_A+r_B >d$ , the circles overlap.
2) If $r_A+r_B< d$ , then no overlap.
3) If $r_A-r_B >d$ , then Circle A contain Circle B.

In our case, since $r_A -r_B >d$ ,
Hence, Circle B is contained in Circle A.

Two circles have the following equations (x -1 )^2+(y -4 )^2= 64 (x -1 )^2+(y -4 )^2= 64 and (x +3 )^2+(y -4 )^2= 9 (x +3 )^2+(y -4 )^2= 9 . 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?