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Circle A has a radius of #3 # and a center at #(1 ,2 )#. Circle B has a radius of #5 # and a center at #(3 ,7 )#. 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
Jun 2, 2018

Answer:

#"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>#

#(3,7)to(3+2,7+1)to(5,8)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)=(1,2)" and "(x_2,y_2)=(5,8)#

#d=sqrt((5-1)^2+(8-2)^2)=sqrt(16+36)=sqrt52~~7.211#

#"sum of radii "=3+5=8#

#"since sum of radii"> d" then circles overlap"#
graph{((x-1)^2+(y-2)^2-9)((x-5)^2+(y-8)^2-25)=0 [-20, 20, -10, 10]}