Let $z$ be a complex number such that $|z|+z=2+i$, where $i=\sqrt{-1}$, then $|z|$ is equal to

Question

Aicharya · Accepted Answer

$\begin{aligned} & |\mathrm{z}|+\mathrm{z}=2+\mathrm{i} \\ & \Rightarrow \sqrt{x^2+y^2}+x+\mathrm{i} y=2+\mathrm{i} \end{aligned}$
Equating real and imaginary parts, we get $\begin{aligned} & \sqrt{x^2+y^2}+x=2 \text { and } y=1 \\ \Rightarrow & \sqrt{x^2+1}=2-x \\ \Rightarrow & x^2+1=4-4 x+x^2 \\ \Rightarrow & 4 x=3 \\ & \Rightarrow x=\frac{3}{4} \\ \therefore \quad & \mathrm{z}=\frac{3}{4}+\mathrm{i} \\ & \Rightarrow|z|=\sqrt{\left(\frac{3}{4}\right)^2+1^2}=\frac{5}{4} \end{aligned}$

Let $z$ be a complex number such that $|z|+z=2+i$, where $i=\sqrt{-1}$, then $|z|$ is equal to

Solution