Circle A has a center at #(2 ,12 )# and an area of #81 pi#. Circle B has a center at #(1 ,3 )# and an area of #36 pi#. Do the circles overlap? If not, what is the shortest distance between them?

1 Answer
Sep 28, 2017

#"circles overlap"#

Explanation:

#"what we have to do here is to "color(blue)"compare"#
#"the distance (d) between the centres of the circles"#
#"with the "color(blue)"sum of the radii"#

#• " if sum of radii ">d" then circles overlap"#

#• " if sum of radii "< d" then no overlap"#

#"to calculate the radii of the circles use"#

#• " area of circle "=pir^2larr" r is the radius"#

#color(blue)"circle A"color(white)(x)pir^2=81pirArrr=9#

#color(blue)"circle B "pir^2=36pirArrr=6#

#"to calculate d use the "color(blue)"distance formula"#

#color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))#

#"here "(x_1,y_1)=(1,3)" and "(x_2,y_2)=(2,12)#

#d=sqrt((2-1)^2+(12-3)^2)=sqrt(1+81)=sqrt82~~ 9.055#

#"sum of radii "=9+6=15#

#"since sum of radii ">d" then circles overlap"#
graph{((x-2)^2+(y-12)^2-81)((x-1)^2+(y-3)^2-36)=0 [-40, 40, -20, 20]}