If #|z| = Max{|z-2|,|z+2|}#, then?

Question

A) #|z+barz| = 1#
B)#z+barz=2^2#
C)#|z+barz| = 2#
D) None of these

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Answer 1 · 2018-03-21T10:02:58

C)

This is equivalent to:

Determine #x,y# such that

#x^2+y^2 = max((x-2)^2+y^2, (x+2)^2+y^2)#

which is equivalent to

#x^2 = max((x-2)^2, (x+2)^2)# or

#x^2=(xpm 2)^2 rArr 0 = pm 4x+4rArr x = pm 1# then

#x^2=max((x-2)^2, (x+2)^2) rArr x = pm 1#

So this gives a two lines set

#{-1,y}# and #{1,y}# or #z_1 = 1+i y# and #z_2 = -1+i y#

then we have

#z_1 + bar z_1 = 2 rArr abs(z_1 + bar z_1)= 2# and
#z_2 + bar z_2 = -2 rArr abs(z_2 + bar z_2)= 2#

so the answer is C)