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"#