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

Question

1129 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-06-28T14:41:44

The circles do not overlap and the minimum distance is #=1.08#

Circle #A#, center #O_A=(2,7)#

The equation of circle #A# is

#(x-2)^2+(y-7)^2=9#

Circle #B#, center #O_B=(6,1)#

The equation of circle #B# is

#(x-6)^2+(y-1)^2=4#

The center of circle #B'# after translation is

#(6,1)+(2,7)=(8,8)#

Circle #B'#, center #O_B'=(8,8)#

The equation of the circle after translation is

#(x-8)^2+(y-8)^2=4#

The distance #O_AO_B'# is

#=sqrt((8-2)^2+(8-7)^2)#

#=sqrt(36+1)#

#=sqrt37#

#=6.08#

This distance is greater than the sum of the radii

#O_AO_B'>r_A+r_B'#

So, the circles do not overlap and the minimum distance is

#=6.08-(2+3)#

#=1.08#
graph{((x-2)^2+(y-7)^2-9)((x-6)^2+(y-1)^2-4)((x-8)^2+(y-8)^2-4)=0 [-7.28, 18.03, -1.57, 11.09]}