<p>If \(\arg\left(\frac{z_1 - z/|z|}{z/|z|}\right) = \frac{\pi}{2}\) and \(\left|\frac{z}{|z|} - z_1\right| = 3\), then \(|z_1|\) equals</p>
Step-by-Step Solution
Key Concept: The expression z/|z| represents the unit vector in the direction of z. The argument condition means the numerator is perpendicular to the denominator (purely imaginary ratio), which constrains z₁ to lie on a specific circle. Combined with the distance condition, we can solve for |z₁|.
<p><strong>Step 1:</strong> Let w = z/|z|, which is a unit complex number (|w| = 1).</p><p><strong>Step 2:</strong> The condition arg((z₁ - w)/w) = π/2 means (z₁ - w)/w is purely imaginary. This gives z₁ - w = ki·w for some real k, so z₁ = w(1 + ki).</p><p><strong>Step 3:</strong> Therefore |z₁| = |w|·|1 + ki| = 1·√(1 + k²) = √(1 + k²).</p><p><strong>Step 4:</strong> From the condition |w - z₁| = 3, we have |w - w(1 + ki)| = 3, which gives |w|·|-ki| = 3, so |k| = 3.</p><p><strong>Step 5:</strong> Thus |z₁| = √(1 + 9) = √10.</p><p>∴ Answer: B</p>
Correct Answer: B