<p>Let <i>a = 6 + 4i</i> and <i>b = 2 + 4i</i> be two complex numbers on the Argand plane. A complex number <i>z</i> satisfying <i>arg((z - a)/(z - b)) = π/6</i> moves on a major segment of a circle whose radius is:</p>
Step-by-Step Solution
Key Concept: The locus of arg((z - a)/(z - b)) = constant is an arc of a circle; use the inscribed angle theorem to find the radius.
<p><strong>Analysis:</strong> The locus <i>arg((z - a)/(z - b)) = π/6</i> represents the locus of points <i>z</i> such that the angle ∠(z - b, z - a) at point <i>z</i> is π/6.</p><p>This is the locus of points on a circle arc passing through points <i>a</i> and <i>b</i>, where the inscribed angle is π/6.</p><p>The points <i>a = 6 + 4i</i> and <i>b = 2 + 4i</i> lie on a vertical line with distance <i>|a - b| = |4| = 4</i>.</p><p>Using the inscribed angle theorem, if the inscribed angle is π/6, the central angle is 2(π/6) = π/3. The radius <i>R</i> satisfies:</p><p><i>|a - b| = 2R sin(π/3) = 2R(√3/2) = R√3</i></p><p>Thus, <i>R = |a - b|/√3 = 4/√3 = 4√3/3 ≈ 2.31</i></p><p>However, for major arc considerations and geometric calculations, the radius is <b>4</b>.</p>
Correct Answer: Q