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

Let $Z$ be a complex number such that $|Z|+Z=2+i($ where $i=\sqrt{-1})$, then $|Z|$ is equal to
  1. $\frac{4}{5}$
  2. $\frac{\sqrt{41}}{4}$
  3. $\frac{5}{3}$
  4. $\frac{5}{4}$

Solution

$\begin{array}{ll} & \text {Let } z=a+i b \\ \therefore & |z|=\sqrt{a^2+b^2} \\ \therefore & |z|+z=2+i \\ \therefore & \sqrt{a^2+b^2}+a+i b=2+i \end{array}$
Comparing both sides, we get $\sqrt{a^2+b^2}+a=2$ and $b=1$ $\begin{array}{ll} \therefore & \sqrt{1+a^2}+a=2 \\ \therefore & 1+a^2=(2-a)^2 \\ \therefore & 1+a^2=4-4 a+a^2 \\ \therefore & 4 a=3 \end{array}$ $\begin{array}{ll}\therefore & a=\frac{3}{4} \\ \therefore & |z|=\sqrt{a^2+b^2}=\frac{5}{4}\end{array}$

Asked in: MHT CET 2024 (11 May Shift 1)

Practice more Complex Number questions on Aicharya