Circle A has a radius of #2 # and a center at #(8 ,3 )#. Circle B has a radius of #3 # and a center at #(3 ,2 )#. If circle B is translated by #<-2 ,6 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

1 Answer
Jun 15, 2018

#"no overlap ",~~3.6#

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 "< -2,6>#

#(3,2)to(3-2,2+6)to(1,8)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)=(8,3)" and "(x_2,y_2)=(1,8)#

#d=sqrt((1-8)^2+(8-3)^2)#

#color(white)(d)=sqrt(49+25)=sqrt74~~8.60#

#"sum of radii "=2+3=5#

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

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

#color(white)(xxxxxxxxxxxxxx)=8.6-5=3.6#
graph{((x-8)^2+(y-3)^2-4)((x-1)^2+(y-8)^2-9)=0 [-20, 20, -10, 10]}