Question

If \( |w| = 2 \), then the set of points \( z = w - \frac{1}{w} \) is contained in or equal to the set of points \( z \) satisfying:

• A. \( {Im}(z) = 0 \)
• B. \( |{Im}(z)| \leq 1 \)
• C. \( |{Re}(z)| \leq 2 \)
• D. \( |z| \leq 3 \)

Hint:

To find the modulus of a complex number expression like \( z = w - \frac{1}{w} \), use polar form and apply the properties of magnitudes. The result will help determine the maximum value.

Answer 1

The Correct Option is D

We are given that \( |w| = 2 \), meaning the modulus of \( w \) is 2. The given expression for \( z \) is: \[ z = w - \frac{1}{w} \] We need to determine the set of points \( z \) satisfies. First, note that: \[ w = 2 \cdot e^{i\theta} \quad {(using polar form of complex numbers, where \( \theta \) is the argument of \( w \))} \] Then, \( \frac{1}{w} \) is the reciprocal of \( w \), which is: \[ \frac{1}{w} = \frac{1}{2} e^{-i\theta} \] Thus, the expression for \( z \) becomes: \[ z = 2e^{i\theta} - \frac{1}{2} e^{-i\theta} \] Now, to find the modulus \( |z| \), we calculate: \[ |z| = \left| 2e^{i\theta} - \frac{1}{2} e^{-i\theta} \right| \] The maximum value of \( |z| \) occurs when \( e^{i\theta} \) and \( e^{-i\theta} \) are aligned such that the magnitude of \( z \) is maximized. By calculation, it turns out that: \[ |z| \leq 3 \] Thus, the correct answer is Option D.