Circle A has a radius of #6 # and a center of #(2 ,5 )#. Circle B has a radius of #3 # and a center of #(1 ,7 )#. If circle B is translated by #<3 ,1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

1 Answer
Jul 2, 2018

Answer:

#"circles overlap"#

Explanation:

#"What we have to do here is compare the distance d "#
#"between the centres of the circles 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 "< 3,1>#

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

#d=sqrt((4-2)^2+(8-5)^2)=sqrt(4+9)=sqrt13~~3.61#

#"sum of radii "=6+3=9#

#"Since sum of radii">d" then circles overlap"#
graph{((x-2)^2+(y-5)^2-36)((x-4)^2+(y-8)^2-9)=0 [-40, 40, -20, 20]}