Question
If $$\left| z \right| = \max \left\{ {\left| {z - 1} \right|,\left| {z + 1} \right|} \right\}$$ then
A.
$$\left| {z + \overline z } \right| = \frac{1}{2}$$
B.
$$ {z + \overline z } = 1$$
C.
$$\left| {z + \overline z } \right| = 1$$
D.
None of these
Answer :
$$\left| {z + \overline z } \right| = 1$$
Solution :
$$\eqalign{
& \left| z \right| = \left| {z - 1} \right| \cr
& \Rightarrow \,\,{\left| z \right|^2} = {\left| {z - 1} \right|^2} \cr
& \Rightarrow \,\,z\overline z = \left( {z - 1} \right)\left( {\overline z - 1} \right) \cr
& \Rightarrow \,\,z\overline z = z\overline z - z - \overline z + 1 \cr
& \therefore \,\,z + \overline z = 1 \cr
& \left| z \right| = \left| {z + 1} \right| \cr
& \Rightarrow \,\,z + \overline z = - 1\left( {{\text{similarly}}} \right) \cr
& \therefore \,\,\left| {z + \overline z } \right| = 1. \cr} $$