If $| z | = M a x {| z - 2 |, | z + 2 |}$ $|z| = Max{|z-2|,|z+2|}$ , then?

Question

A) $| z + ¯ z | = 1$ $|z+barz| = 1$
B) $z + ¯ z = 2^{2}$ $z+barz=2^2$
C) $| z + ¯ z | = 2$ $|z+barz| = 2$
D) None of these

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Answer 1 · 2018-03-21T10:02:58

C)

This is equivalent to:

Determine $x, y$ $x,y$ 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)