Circle A has a center at #(2 ,5 )# and an area of #16 pi#. Circle B has a center at #(8 ,1 )# and an area of #3 pi#. Do the circles overlap? If not, what is the shortest distance between them?

1 Answer
May 22, 2018

#"no overlap "~~1.479#

Explanation:

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

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

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

#"to find the radii use area (A) of circle formula"#

#•color(white)(x)A=pir^2#

#"circle A "to pir^2=16pirArrr=4#

#"circle B "to pir^2=3pirArrr=sqrt3#

#"calculate d using the "color(blue)"distance formula"#

#•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

#"let "(x_1,y_1)=(2,5)" and "(x_2,y_2)=(8,1)#

#d=sqrt((8-2)^2+(1-5)^2)=sqrt(36+16)=sqrt52~~7.211#

#"sum of radii "=4+sqrt3~~5.732#

#"since sum of radii"< d" then no overlap"#

#"minimum distance "=d-" sum of radii"#

#color(white)(xxxxxxxxxxxxxx)=7.211-5.732=1.479#
graph{((x-2)^2+(y-5)^2-16)((x-8)^2+(y-1)^2-3)=0 [-10, 10, -5, 5]}