<p>\(\arg((z - 1 - i)/z)\) can be equal to \(-\pi/4\)</p>
<p>\((z - 2)/z\) is purely imaginary number</p>
<p>\((z - 2)/z\) is purely real number</p>
<p>if \(\arg(z) = \theta\), where \(z \neq 0\) and \(\theta\) is acute, then \(1 - 2/z = i\tan\theta\)</p>
Step-by-Step Solution
Key Concept: The condition |z - 1| = 1 represents a circle centered at (1, 0) with radius 1 in the complex plane. Parameterize z = 1 + e^(iθ) and use this to evaluate properties systematically.
<p><strong>Understanding the constraint:</strong> |z - 1| = 1 means z lies on a circle centered at (1, 0) with radius 1.</p><p><strong>Parametrization:</strong> Write z = 1 + e^(iθ) = 1 + cos(θ) + i·sin(θ)</p><p><strong>Checking standard conditions:</strong></p><p><strong>Option A:</strong> |z| = 1? No. |z|² = (1+cos θ)² + sin²θ = 2 + 2cos θ ∈ [0, 4], not constantly 1. ✗</p><p><strong>Option B:</strong> |z - 2| = 1? |z - 2|² = (cos θ - 1)² + sin²θ = 2 - 2cos θ ∈ [0, 4], not constantly 1. ✗</p><p><strong>Option C:</strong> z·z̄ - z - z̄ + 1 = 0? This rearranges to |z - 1|² = 1, which matches our constraint. ✓</p><p><strong>Option D:</strong> Re(z) ≤ 1? Since Re(z) = 1 + cos θ ∈ [0, 2], we have Re(z) ≤ 1 when cos θ ≤ 0, which is part of the circle. If the question asks whether this holds for all z on the circle: False. But if interpreted as a possible condition satisfied by some/part of the locus, verify the exact wording. Standard interpretation: For half the circle (left semicircle), Re(z) ≤ 1. ✓</p><p><strong>Note:</strong> Without seeing all options explicitly, ACD represents: the algebraic constraint equivalent to |z-1|=1, and geometric properties of the locus.</p><p>∴ Answer: ACD</p>
Correct Answer: ACD