Circle A has a radius of #4 # and a center at #(8 ,2 )#. Circle B has a radius of #3 # and a center at #(4 ,5 )#. If circle B is translated by #<-3 ,4 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

1 Answer
Jul 31, 2018

#"no overlap "~~2.9#

Explanation:

#"What we have to do here is to compare the distance (d)"#
#"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 a translation "< -3, 4>#

#(4,5)to(4-3,5+4)to(1,9)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)=(8,2)" and "(x_2,y_2)=(1,9)#

#d=sqrt((1-8)^2+(9-2)^2)=sqrt(49+49)=sqrt98~~9.9#

#"sum of radii "=4+3=7#

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

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

#color(white)(xxxxxxxxxxxxx)=9.9-7=2.9#
graph{((x-8)^2+(y-2)^2-16)((x-1)^2+(y-9)^2-9)=0 [-20, 20, -10, 10]}