Circle A has a radius of #2 # and a center of #(2 ,5 )#. Circle B has a radius of #3 # and a center of #(3 ,8 )#. If circle B is translated by #<4 ,-1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

1 Answer
Jun 16, 2018

#"no overlap ",~~0.385#

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 "<4,-1>#

#(3,8)to(3+4,8-1)to(7,7)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)=(2,5)" and "(x_2,y_2)=(7,7)#

#d=sqrt((7-2)^2+(7-5)^2)=sqrt(25+4)=sqrt29~~5.385#

#"sum of radii "=2+3=5#

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

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

#color(white)(xxxxxxxxxxxxx)=5.385-5=0.385#
graph{((x-2)^2+(y-5)^2-4)((x-7)^2+(y-7)^2-9)=0 [-20, 20, -10, 10]}