Question

$P\left(z\right)$ be a variable point in the Argand plane such that $|z|=minimum\{|z-1|,|z+1|\}$, then $z+\overline{z}$ will be equal to

• A. $-1$ or $1$
• B. $1$ but not equal to $-1$
• C. $-1$ but not equal to $1$
• D. None of these

Answer 1

The correct option is A $-1$ or $1$

When $|z-1|<|z+1|$ (or $x>0)$
$|z|=|z-1|$
$\Rightarrow x^2+y^2=(x-1{)}^2+y^2$
$\Rightarrow x=1/2$
$\Rightarrow z+\overline{z}=1$
When $|z-1|>|z+1|$ (or $x<0)$,
$|z|=|z+1|$
$\Rightarrow x^2+y^2=(x+1{)}^2+y^2$
$\Rightarrow x=-1/2$
$\Rightarrow z+\overline{z}=-1$