Two circles have the following equations $(x -1 )^2+(y -4 )^2= 36$ and $(x +5 )^2+(y -2 )^2= 81$ . 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 · 2017-03-04T19:05:42

$"One circle contain the other."$

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$"center coordinates : " O(x,y)$

$"radius : "r$

$"equation of circle : "(x-a)^2+(y-b)^2=r^2$

$"we can write as:"$

$" for the blue circle : " a=1 , b=4 , r_("blue")=sqrt(36)=6$

$"and for the red circle : " a=-5 , b=2 , r_("red")=sqrt(81)=9$

$r_("blue")+r_("red")=6+9=15$

$"now let us find distance between "O_1 " and "O_2$

$"let distance between "O_1 " and "O_2 " be 'l'"$

$l=sqrt((4-2)^2+(5+1)^2)$

$l=sqrt(2^2+6^2)=sqrt(4+36)=sqrt(40)$

$l=6.32$

$"if l >"r_("blue")+r_("red")" then no overlap"$

$"if l<"r_("blue")+r_("red")" then overlap"$

$6.32 <15" overlap"$

Two circles have the following equations (x -1 )^2+(y -4 )^2= 36 (x -1 )^2+(y -4 )^2= 36 and (x +5 )^2+(y -2 )^2= 81 (x +5 )^2+(y -2 )^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?