Circle A has a radius of #4 # and a center of #(6 ,2 )#. Circle B has a radius of #2 # and a center of #(5 ,3 )#. If circle B is translated by #<-2 ,2 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Answer 1 · 2016-04-06T10:05:13

The degree of overlap is

#6-3sqrt(2)~~1.757# to 3 decimal places

For circle A
Let the centre be #C_a -> (6,2)#
Let radius be #R_a -> 4#

For circle B
Let the centre be #C_b->(5,3)#
Let the radius be #R_b->2#

Let distance between centres be #D#

#C_b # translated by #<-2,2 >#

#=> C_b ->(5-2,3+2)#

#=> C_b ->(3,5)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine final distance between centres")#

#D=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

#D=sqrt((3-6)^2+(5-2)^2)#

#D=sqrt((-3)^2+(3)^2)#

#color(blue)(D=sqrt(18) = 3sqrt(2))#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine if circles overlap")#

For this to be true we need #D < R_a+R_b#

#R_a+R_B = 4+2=6#

#3sqrt(2)<6# so they do overlap

The degree of overlap is

#6-3sqrt(2)~~1.757# to 3 decimal places