Circle A has a radius of #2 # and a center at #(3 ,6 )#. Circle B has a radius of #4 # and a center at #(2 ,3 )#. If circle B is translated by #<-2 ,1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

1 Answer
Jul 15, 2018

#"circles overlap"#

Explanation:

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

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

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

#"Before calculating d we require to find the new"#
#"centre of B under the given translation"#

#"under the translation "< -2,1 >#

#B(2,3)to(2-2,3+1)to(0,4)larrcolor(red)"new centre of B"#

#"to calculate d use 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)=(3,6)" and "(x_2,y_2)=(0,4)#

#d=sqrt((0-3)^2+(4-6)^2)=sqrt(9+4)=sqrt13~~3.61#

#"sum of radii "=2+4=6#

#"Since sum of radii"> d" then circles overlap"#
graph{((x-3)^2+(y-6)^2-4)((x-0)^2+(y-4)^2-16)=0 [-20, 20, -10, 10]}