Question

Let $z$ be a complex number such that $|z|+z=3+i$ (where $i=\sqrt{-1})$. Then $|z|$ is equal to :

• A. $\frac{5}{3}$
• B. $\frac{5}{4}$
• C. $\frac{\sqrt{34}}{3}$
• D. $\frac{\sqrt{41}}{4}$

Answer 1

The correct option is A $\frac{5}{3}$

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

Thus, $|z|=\sqrt{x^2+y^2}$
$=\sqrt{{\left(\frac{4}{3}\right)}^2+(1{)}^2}$
$=\frac{5}{3}$