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 $5 / 2$
(2014 Main)
(b) lies in the interval $(1,2)$
(c) is strictly greater than $5 / 2$
(d) is strictly greater than $3 / 2$ but less than $5 / 2$
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Answer:
Correct Answer: 7. (b)
Solution:
- $|z| \geq 2$ 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 $(-2,0)$
$$ =\sqrt{-2+\frac{1}{2}^{2}+0}=\frac{3}{2}=\sqrt{\frac{-1}{2}+2^{2}+0}=\frac{3}{2} $$
Geometrically Min $\left|z+\frac{1}{2}\right|=A D$
Hence, minimum value of $\left|z+\frac{1}{2}\right|$ lies in the interval $(1,2)$.
