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

## A) $| z + \overline{z} | = 1$ B)$z + \overline{z} = {2}^{2}$ C)$| z + \overline{z} | = 2$ D) None of these

Mar 21, 2018

C)

#### Explanation:

This is equivalent to:

Determine $x , y$ such that

${x}^{2} + {y}^{2} = \max \left({\left(x - 2\right)}^{2} + {y}^{2} , {\left(x + 2\right)}^{2} + {y}^{2}\right)$

which is equivalent to

${x}^{2} = \max \left({\left(x - 2\right)}^{2} , {\left(x + 2\right)}^{2}\right)$ or

${x}^{2} = {\left(x \pm 2\right)}^{2} \Rightarrow 0 = \pm 4 x + 4 \Rightarrow x = \pm 1$ then

${x}^{2} = \max \left({\left(x - 2\right)}^{2} , {\left(x + 2\right)}^{2}\right) \Rightarrow x = \pm 1$

So this gives a two lines set

$\left\{- 1 , y\right\}$ and $\left\{1 , y\right\}$ or ${z}_{1} = 1 + i y$ and ${z}_{2} = - 1 + i y$

then we have

${z}_{1} + {\overline{z}}_{1} = 2 \Rightarrow \left\mid {z}_{1} + {\overline{z}}_{1} \right\mid = 2$ and
${z}_{2} + {\overline{z}}_{2} = - 2 \Rightarrow \left\mid {z}_{2} + {\overline{z}}_{2} \right\mid = 2$