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

The correct option is B

$\frac{3}{2}-2i$

Explanation for the correct option

$\left|z\right|-z=1+2i$

Let $z=x+iy$

$\left|z\right|=\sqrt{x^2+y^2}$

$$\begin{gathered} \left|z\right|-z=1+2i \\ \Rightarrow \sqrt{x^2+y^2}-(x+iy)=1+2i \\ \Rightarrow \sqrt{x^2+y^2}=1+x+i(2+y) \end{gathered}$$

Comparing real and imaginary parts of both sides

$$\begin{gathered} \Rightarrow 2+y=0 \\ \Rightarrow y=-2 \end{gathered}$$

$1+x=\sqrt{x^2+y^2}$

Squaring both sides

$$\begin{gathered} \Rightarrow {\left(1+x\right)}^2=x^2+y^2 \\ \Rightarrow 1+x^2+2x=x^2+y^2 \\ \Rightarrow y^2=1+2x \end{gathered}$$

Put $y=-2$

$$\begin{gathered} \Rightarrow {(-2)}^2=1+2x \\ \Rightarrow \frac{3}{2}=x \end{gathered}$$

Therefore, solution $=\frac{3}{2}-2i$

Hence, option B is correct.