Question

The complex number $z$ satisfying the equation $|z|=z+1+2i$ is

• A. $\frac{3}{2}-2i$
• B. $\frac{3}{2}+2i$
• C. $\frac{2}{3}-2i$
• D. $\frac{3}{2}-\frac{i}{2}$

Answer 1

The correct option is A $\frac{3}{2}-2i$

Let $z=x+iy$
Given that $|z|=z+1+2i$
$\Rightarrow \sqrt{x^2+y^2}=x+iy+1+2i$
$\Rightarrow \sqrt{x^2+y^2}=(x+1)+(2+y)i$
On comparing real and imaginary parts,
$\sqrt{x^2+y^2}=x+1$
and $0=2+y\Rightarrow y=-2$
$\Rightarrow \sqrt{x^2+4}=x+1$
$\Rightarrow x^2+4=x^2+2x+1$
$\Rightarrow 2x=3$
$\Rightarrow x=\frac{3}{2}$
$\therefore z=x+iy=\frac{3}{2}-2i$