If z is a complex number such that left| z right| ge 2 , then minimum value of left| {z + frac

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4-1.Complex numbers

medium

If $z$ is a complex number such that $\left| z \right| \ge 2$ , then the minimum value of $\left| {z + \frac{1}{2}} \right|$:

is strictly greater than $\frac{5}{2}$

is strictly greater than $\;\frac{3}{2}$ but less than $\frac{5}{2}$

is equal to $\frac{5}{2}$

lie in the interval $(1,2)$

(JEE MAIN-2014)

Solution

$|z| \geq 2$ is the region on or outside circle whose centre is $(0,0)$ and the radius is $2$ .

Minimum $\left|z+\frac{1}{2}\right|$ is distance of $z$, which lies on circle $|z|=2$ from $\left(-\frac{1}{2}, 0\right)$ therefore, minimum $\left|z+\frac{1}{2}\right|=$ Distance of $\left(-\frac{1}{2}, 0\right)$ from $(-2,0)$

$=\sqrt{\left(-2+\frac{1}{2}\right)^{2}+0}=\frac{3}{2}=\sqrt{\left(\frac{-1}{2}+2\right)^{2}+0}=\frac{3}{2}$

Hence, minimum value of $\left|z+\frac{1}{2}\right|$ lies in the interval $(1,2)$

Standard 11

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If $z$ is a complex number such that $\left| z \right| \ge 2$ , then the minimum value of $\left| {z + \frac{1}{2}} \right|$:

Solution

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