Circle A has a radius of #2 # and a center at #(3 ,1 )#. Circle B has a radius of #4 # and a center at #(8 ,3 )#. If circle B is translated by #<-2 ,4 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Jim G. Share
Mar 1, 2018

Answer:

#"no overlap "~~0.71" units"#

Explanation:

#"What we have to do here is to "color(blue)"compare ""the"#
#"distance (d) between the centres with the "color(blue)"sum of 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 centre of B"#
#"under the given translation"#

#"under the translation "<-2,4>#

#(8,3)to(8-2,3+4)to(6,7)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)=(3,1)" and "(x_2,y_2)=(6,7)#

#d=sqrt((6-3)^2+(7-1)^2)=sqrt(9+36)=sqrt45~~6.71#

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

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

#"min. distance "=d-" sum of radii"#

#color(white)("min. distance ")=6.71-6=0.71#
graph{((x-3)^2+(y-1)^2-4)((x-6)^2+(y-7)^2-16)=0 [-20, 20, -10, 10]}

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