Question

The solution of equation $\left|z\right|-z=1+2i$ is

• A. $\frac{3}{2}+2i$
• B. $\frac{3}{2}-2i$
• C. $3-2i$
• D. None of these

Answer 1

Answer: (B)

Let $z=x+iy$
Given, $\left|z\right|-z=1+2i$
$\Rightarrow \sqrt{x^2+y^2}-\left(x+iy\right)=1+2i$
$\Rightarrow \sqrt{x^2+y^2}-x=1,y=-2$
$\Rightarrow \sqrt{x^2+4}-x=1$
$\Rightarrow x^2+4={\left(1+x\right)}^2$
$\Rightarrow 2x=3\Rightarrow x=\frac{3}{2}$
$\therefore z=\frac{3}{2}-2i$