Question

The value of $k$ for which the inequality $\left|Re\right(z\left)\right|+\left|lm\right(z\left)\right|\le k\left|z\right|$ is true for all $z\in C$, is

• A. $2$
• B. $\sqrt{2}$
• C. $1$
• D. None of these

Answer 1

Answer: (B)

$\sqrt{2}$

Let $z=r(\cos \theta +i\sin \theta )$
then
$\left|z\right|=r;arg\left(z\right)=0$
Now, $\left|Re\right(z\left)\right|+\left|Im\right(z\left)\right|\le \left|r\cos \theta \right|+\left|r\sin \theta \right|$
$\Rightarrow \left|Re\right(z\left)\right|+\left|Im\right(z\left)\right|=r(\left|\cos \theta \right|+\left|\sin \theta \right|)$
$\Rightarrow {\{\left|Re\right(z\left)\right|+\left|Im\right(z\left)\right|\}}^2=r^2\left\{|1+r\sin \theta |\right\}$
$\Rightarrow {\{\left|Re\right(z\left)\right|+\left|Im\right(z\left)\right|\}}^2\le r^2(1+1)$
$\Rightarrow \left|Re\right(z\left)\right|+\left|Im\right(z\left)\right|\le \sqrt{2}r$
$\Rightarrow \left|Re\right(z\left)\right|+\left|Im\right(z\left)\right|\le \sqrt{2}\left|z\right|$
$\therefore k=\sqrt{2}$