If z is a complex number such that |z| ge 2, then the minimum value of |z+(1/2)|

Question

If z is a complex number such that |z| ge 2, then the minimum value of |z+(1/2)| (A) is strictly greater than (5/2) (B) is strictly greater than (3/2) but less 5 than (5/2) (C) is equal to (5/2) (D) lies in the interval (1, 2)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/9067deef84ef676cf6432014fcfd18d1-.png" /> |z+(1/2)| So, | z|-(1/2)| ≤|z+(1/2)| ⇒ |z+(1/2)| ≥|2-(1/2)| ⇒ |z min .=(3/2)|

Q. If z is a complex number such that $∣ z ∣ \geq 2,$ then the minimum value of $∣ ∣ z + \frac{1}{2} ∣ ∣$

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

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

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

lies in the interval (1, 2)

$∣ ∣ z + \frac{1}{2} ∣ ∣$
So, $∣ z ∣ ∣ - \frac{1}{2} ∣ ∣ \leq ∣ ∣ z + \frac{1}{2} ∣ ∣$
$\Rightarrow ∣ ∣ z + \frac{1}{2} ∣ ∣ \geq ∣ ∣ 2 - \frac{1}{2} ∣ ∣$
$\Rightarrow ∣ ∣ z_{m i n .} = \frac{3}{2} ∣ ∣$

Q. If z is a complex number such that ∣z∣≥2, then the minimum value of ∣∣​z+21​∣∣​

is strictly greater than 25​

is strictly greater than 23​ but less 5 than 25​

is equal to 25​