Question

If \( \left| z + \frac{2}{z} \right| = 2 \), then the minimum value of \( |z| \) is

• A. \( 1 + \sqrt{2} \)
• B. \( 1 + 2\sqrt{2} \)
• C. \( 3\sqrt{3} + 1 \)
• D. \( 1 - \sqrt{3} \)

Hint:

For expressions like \( z + \frac{1}{z} \), always substitute \( |z| = r \).

Answer 1

The Correct Option is A

Concept: Use triangle inequality: \[ \left| z + \frac{2}{z} \right| \ge \left||z| - \frac{2}{|z|}\right| \]

Step 1: Let \( |z| = r \).
\[ \left| r + \frac{2}{r} \right| = 2 \]

Step 2: Minimize expression.
\[ r + \frac{2}{r} = 2 \Rightarrow r^2 - 2r + 2 = 0 \]

Step 3: Solve.
\[ r = 1 \pm \sqrt{1 - 2} \Rightarrow r = 1 + \sqrt{2} \]