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