Question

If $z$ is a complex number such that $|z| \geq 1$, then the minimum value of $\left| z + \frac{1}{2}(3 + 4i) \right|$is:

• A. $\frac{5}{2}$
• B. 2
• C. 3
• D. $\frac{3}{2}$
• E. $\frac{3}{2}$

Answer 1

Answer: (E)

$\frac{3}{2}$

Answer 2

Given:

$\text{Minimize } \left| z + \frac{1}{2}(3 + 4i) \right| \quad \text{subject to } |z| \geq 1$

Let:

$w = \frac{1}{2}(3 + 4i)$ $w = \left(\frac{3}{2}, 2\right)$

We need to find the minimum value of:

$|z + w| = \left| z + \left(\frac{3}{2} + 2i\right) \right|$

subject to $|z| \geq 1$.

The minimum distance from a point $w$ to any point $z$ on or outside the unit circle occurs when $z$ lies on the boundary of the circle.

Thus, we find the minimum distance between $w$ and the unit circle centered at the origin.

Distance Calculation

The distance of $w = \left(\frac{3}{2}, 2\right)$ from the origin is given by:

$|w| = \sqrt{\left(\frac{3}{2}\right)^2 + 2^2} = \sqrt{\frac{9}{4} + 4} = \sqrt{\frac{25}{4}} = \frac{5}{2}$

Since $|z| \geq 1$, the minimum value of $|z + w|$ is obtained when $z$ lies on the circle of radius 1, making the minimum value:

$|w| - 1 = \frac{5}{2} - 1 = \frac{3}{2}$

Conclusion: The minimum value of $\left| z + \frac{1}{2}(3 + 4i) \right|$ is $\frac{3}{2}$.