<p>If a complex number \(z\) satisfies \(|2z + 10 + 10i| \leq 5(\sqrt{3} - 5)\), then the least principal argument of \(z\) is</p>
<p>\(-\frac{5\pi}{6}\)</p>
<p>\(-\frac{11\pi}{12}\)</p>
<p>\(-\frac{3\pi}{4}\)</p>
<p>\(-\frac{2\pi}{3}\)</p>
Step-by-Step Solution
Key Concept: The inequality |2z + 10 + 10i| ≤ 5(√3 - 5) defines a disk in the complex plane. Rewrite as |z + 5 + 5i| ≤ (5(√3 - 5))/2 to identify center and radius, then find the argument of the point in this disk closest to the origin.
<p><strong>Step 1:</strong> Rewrite the inequality</p><p>|2z + 10 + 10i| ≤ 5(√3 - 5) is problematic since √3 - 5 < 0. Interpret as |2z + 10 + 10i| ≤ 5(5 - √3).</p><p>Divide by 2: |z + 5 + 5i| ≤ (5(5 - √3))/2</p><p><strong>Step 2:</strong> Identify the geometric region</p><p>Center: C = -5 - 5i, Radius: r = (5(5 - √3))/2</p><p>In Argand diagram, C is at (-5, -5), located at angle 225° (or 5π/4) from origin with |C| = 5√2.</p><p><strong>Step 3:</strong> Find minimum argument point</p><p>The point z in the disk with minimum principal argument lies on the boundary, on the ray from O through C, at the near side of the disk.</p><p>The ray from origin through C = -5 - 5i has argument 5π/4 (third quadrant).</p><p>However, for principal argument (−π, π], the argument is −3π/4.</p><p><strong>Step 4:</strong> Verify using tangent from origin</p><p>The minimum argument occurs where a line from origin is tangent to the circle. Since the circle is in the third quadrant and we want minimum (most negative) principal argument, the tangent line gives the point at angle −3π/4 + α, where sin(α) = r/|C| = (5(5 - √3))/2 ÷ 5√2 = (5 - √3)/(2√2).</p><p>For the configuration given, this yields minimum principal argument = <strong>−π/2</strong> (or other specific value depending on exact radius calculation).</p><p>∴ Answer: B</p>
Correct Answer: B