Circle A has a radius of #3 # and a center of #(5 ,4 )#. Circle B has a radius of #1 # and a center of #(7 ,2 )#. If circle B is translated by #<3 ,-5 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Jim G. Share
Feb 16, 2018

Answer:

#"no overlap ",~~4.602#

Explanation:

#"what we have to do here is "color(blue)"compare"" the distance (d)"#
#"to the "color(blue)"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 centre of "#
#"B under the given translation"#

#"under the translation "<3,-5>#

#(7,2)to(7+3,2-5)to(10,-3)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)=(5,4)" and "(x_2,y_2)=(10,-3)#

#d=sqrt((10-5)^2+(-3-4)^2)=sqrt(25+49)~~8.062#

#"sum of radii "=3+1=4#

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

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

#color(white)(xxxxxxxxxx)=8.062-4=4.062#
graph{((x-5)^2+(y-4)^2-9)((x-10)^2+(y+3)^2-1)=0 [-20, 20, -10, 10]}

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