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If $z$ is a complex number with $|z| \geq 5$. Then the least value of $\left|z+\frac{2}{z}\right|$ is
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Verified Answer
The correct answer is:
$\frac{23}{5}$
We have given, $|z| \geq 5$
$$
\begin{aligned}
& \text { Now, }\left|z+\frac{2}{z}\right| \geq|z|-\left|\frac{2}{z}\right|=|| z\left|+\frac{2}{z}\right| \\
& =|| z\left|+2\left(\frac{-1}{|z|}\right)\right| \geq\left|5-\frac{2}{5}\right| \\
& \therefore\left|z+\frac{2}{z}\right| \geq \frac{23}{5}
\end{aligned}
$$
Thus, the least value of $|\quad \underline{2}|$ is $\frac{23}{}$
$$
\begin{aligned}
& \text { Now, }\left|z+\frac{2}{z}\right| \geq|z|-\left|\frac{2}{z}\right|=|| z\left|+\frac{2}{z}\right| \\
& =|| z\left|+2\left(\frac{-1}{|z|}\right)\right| \geq\left|5-\frac{2}{5}\right| \\
& \therefore\left|z+\frac{2}{z}\right| \geq \frac{23}{5}
\end{aligned}
$$
Thus, the least value of $|\quad \underline{2}|$ is $\frac{23}{}$
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