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Circle A has a radius of #5 # and a center of #(6 ,1 )#. Circle B has a radius of #1 # and a center of #(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
Jan 26, 2018

Answer:

#"no overlap"#

Explanation:

What we have to do here is #color(blue)"compare "# thedistance (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 we can calculate d we require the 'new' centre of"#
#"circle B"#

#"under a translation of "<-3,4>#

#(4,5)to(4-3,5+4)to(1,9)larrcolor(red)"new centre of B"#

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

#color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))#

#"let "(x_1,y_1)=(6,1)" and "(x_2,y_2)=(1,9)#

#d=sqrt((1-6)^2+(9-1)^2)=sqrt(25+64)~~9.43#

#"sum of radii "=5+1=6#

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

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

#color(white)("minimum distance ")=9.43-6=3.43#
graph{((x-6)^2+(y-1)^2-25)((x-1)^2+(y-9)^2-1)=0 [-20, 20, -10, 10]}