Question

Let the curve \((\)\( z(1 + i) + \overline{z(1 - i)} = 4\), \(\, z \in \mathbb{C}\) \()\), divide the region \(|z - 3| \leq 1\) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha - \beta| \) equals:

• A. \( 1 + \frac{\pi}{4} \)
• B. \( 1 + \frac{\pi}{2} \)
• C. \( 1 + \frac{\pi}{3} \)
• D. \( 1 + \frac{\pi}{6} \)

Hint:

In geometry problems involving complex numbers and areas: - Convert the given complex equation to geometric terms (e.g., lines, circles). - Use geometric interpretations and symmetry to calculate areas, especially when the curve divides a region into two parts.

Answer 1

The Correct Option is A

The equation represents a line in the complex plane that partitions the disk \( |z - 3| \leq 1 \) into two sections. Leveraging the geometric characteristics of the circle and line, the areas of these two regions can be calculated, yielding:\[|\alpha - \beta| = 1 + \frac{\pi}{4}.\]