<p>Given that the two curves \(\arg(z) = \pi/6\) and \(|z - 2\sqrt{3}\, i| = r\) intersect in two distinct points, then</p>
<p>(1) [<em>r</em>] ≠ 2, where [.] represents greatest integer</p>
<p>(2) 0 < <em>r</em> < 3</p>
<p>(3) <em>r</em> = 6</p>
<p>(4) 3 < <em>r</em> < 2√3</p>
Step-by-Step Solution
Key Concept: The curve arg(z) = π/6 is a ray from origin at angle π/6, and |z - 2√3i| = r is a circle centered at (0, 2√3). For two distinct intersections, the perpendicular distance from center to the ray must be less than radius r, and r must not equal this distance.
<p><strong>Step 1:</strong> The curve arg(z) = π/6 represents a ray from origin: z = t(cos(π/6) + i·sin(π/6)) = t(√3/2 + i/2) for t ≥ 0.</p><p><strong>Step 2:</strong> The circle |z - 2√3i| = r has center C = (0, 2√3) and radius r. The ray passes through origin O(0,0).</p><p><strong>Step 3:</strong> Find perpendicular distance from center C(0, 2√3) to the ray. The ray direction is (√3/2, 1/2). Using point-to-line distance: d = |2√3·sin(π/6)| / 1 = 2√3·(1/2) = √3.</p><p><strong>Step 4:</strong> For two distinct intersection points, we need r > √3 (circle radius exceeds perpendicular distance to ray).</p><p><strong>Step 5:</strong> Also, the intersection points must be on the ray (t ≥ 0, not on the opposite direction). The closest point on ray to C is at distance √3, confirming: <strong>r > √3</strong>.</p><p>∴ Answer: r > √3</p>
Correct Answer: 1