Question

If $z$ be a complex number satisfying $\left|z\right|+2\overline{z}=z$, then complete solution set of $z$ lies on-

• A. the positive real axis including orgin
• B. the positive real axis
• C. the negative imaginary axis including origin
• D. the negative real axis including origin

Answer 1

The correct option is D the negative real axis including origin

Here $\left|z\right|+2\overline{z}=z.........\left(i\right)$
Taking conjugate
$\left|z\right|+2z=\overline{z}........\left(ii\right)$
$\Rightarrow$ subtracting equation 1 and 2 we get,
$\Rightarrow$ $z=\overline{z}$ $\Rightarrow$ $z\in R$
$\Rightarrow$ $\left|z\right|+z=0$ $\Rightarrow$ $z\le 0$
$\Rightarrow$ $z$ lies on negative real axis including origin.