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If $z$ is a complex number such that $\left| z \right| \geqslant 2$ then the minimum value of $\left| {z + \frac{1}{2}} \right|$ is
A. Is strictly greater than $\frac{5}{2}$
B. Is strictly greater than $\frac{3}{2}$ but less than $\frac{5}{2}$
C. Is equal to $\frac{5}{2}$
D. Lies in the interval $(1,2)$

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Last updated date: 18th Jun 2024
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Answer
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Hint: For solving this particular question we have to analysis that $\left| z \right|\geqslant 2$ is the region on or outside the circle whose centre is $(0,0)$and the radius is two. And minimum $\left| {z + \frac{1}{z}} \right|$ is equal to distance between $\left( { - \frac{1}{2},0} \right)$ to $(0,0)$ .

Complete solution step by step:
It is given that $z$ is a complex number such that $\left| z \right| \geqslant 2$ ,
$\left| z \right| \geqslant 2$ is the region on or outside the circle whose centre is $(0,0)$ and the radius is two.
Minimum $\left| {z + \frac{1}{z}} \right|$ is distance of $z$ which lies on the circle $\left| z \right|
= 2$ from $\left( { - \frac{1}{2},0} \right)$ ,
Thus minimum $\left| {z + \frac{1}{z}} \right|$ is equal to distance between $\left( { - \frac{1}{2},0} \right)$ to $(0,0)$ .
$\begin{gathered}
= \sqrt {{{\left( { - \frac{1}{2} + 2} \right)}^2} + {{(0 - 0)}^2}} \\
= \sqrt {{{\left( { - \frac{1}{2} + 2} \right)}^2}} \\
\end{gathered} $
Now apply radial rule that is , $\sqrt[n]{{{a^n}}} = a$ where $a \geqslant 0$ ,
Therefore , we will get ,
$\begin{gathered}
= - \frac{1}{2} + 2 \\
= \frac{{ - 1 + 4}}{2} \\
= \frac{3}{2} \\
\end{gathered} $
Hence , option B is the correct option.

Additional information: As we know that $z = x + yi$ , which is the representation of the complex number. And $z = x - yi$ , is the conjugate of the complex number.
Now multiplication of the complex number with the conjugate of the complex number we get magnitude which represents the distance of the complex number from the origin. we know that ${\left| {{z_1}} \right|^2} = {z_1}\overline {{z_1}} $ , multiplication of the complex number with the conjugate of the complex number we get magnitude which represents the distance of the complex number from the origin.

Note: If $z = x + yi$ be any complex number then modulus of $z$ is represented as $\left| z \right|$ and is equal to $\sqrt {{x^2} + {y^2}} $ . Here $\left| z \right| \geqslant 2$ is the region on or outside the circle whose centre is $(0,0)$and the radius is two.