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)

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 \( \frac{3}{2} \).