Complex Numbers 2 Question 7

7.

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

(a) is equal to $5 / 2$

(b) lies in the interval $(1,2)$

(c) is strictly greater than $5 / 2$

(d) is strictly greater than $3 / 2$ but less than $5 / 2$

(2014 Main)

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Answer:

Correct Answer: 7. (b)

Solution:

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

Minimum $\left|z+\frac{1}{2}\right|$ is distance of $z$, which lie on circle $|z|=2$ from $(-1 / 2,0)$.

$\therefore$ Minimum $\left|z+\frac{1}{2}\right|=$ Distance of $-\frac{1}{2}, 0$ from $(-2,0)$

$ =\sqrt{-2+\frac{1}{2}^{2}+0}=\frac{3}{2}=\sqrt{\frac{-1}{2}+2^{2}+0}=\frac{3}{2} $

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Geometrically Min $\left|z+\frac{1}{2}\right|=A D$

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



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