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

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What we have to do here is

#color(blue)"compare"# the distance (d) between the centres of the circles 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"#

#"under the given translation"#

#"under a translation "<-1,2>#

#(7,3)to(7-1,3+2)to(6,5)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)=(2,7)" and "(x_2,y_2)=(6,5)#

#d=sqrt((6-2)^2+(5-7)^2)=sqrt(16+4)=sqrt20~~4.47#

#"sum of radii "=5+4=9#

#"since sum of radii">d" then circles overlap"#

graph{((x-2)^2+(y-7)^2-25)((x-6)^2+(y-5)^2-16)=0 [-20, 20, -10, 10]}

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