Question

If complex number $z$ satisfies $|z|+z=2+i$, then $z$ is

• A. $z=\frac{3}{4}+2i$
• B. $z=\frac{3}{4}+i$
• C. $z=1+i$
• D. $z=\frac{3}{5}+i$

Answer 1

The correct option is B $z=\frac{3}{4}+i$

Let $z=x+iy$
$\Rightarrow |z|+z=(2+i)$
$\Rightarrow \sqrt{x^2+y^2}+x+iy=(2+i)$
By comparing real and imaginary part, we get
$\sqrt{x^2+y^2}+x=2$ and $y=1$
$\Rightarrow (2-x{)}^2=x^2+y^2$
$\Rightarrow (2-x{)}^2=x^2+1$
$\Rightarrow x=\frac{3}{4}$
$\therefore y=1,x=\frac{3}{4}$

Hence, $z=\frac{3}{4}+i$