<p>If <span>z</span> and <span>\bar{z}</span> represent adjacent vertices of a regular polygon of <span>n</span> sides with centre at origin and if <span>\frac{\text{Im}(z)}{\text{Re}(z)} = \sqrt{2} - 1</span>, the value of <span>n</span> is equal to</p>
Step-by-Step Solution
Key Concept: If z and z̄ are adjacent vertices of a regular n-sided polygon centered at origin, the angle between them is 2π/n. Since z̄ is the complex conjugate of z, they are reflections across the real axis, so the angle from z to z̄ measured at origin is actually π/n (half the central angle between adjacent vertices when considering the conjugate relationship).
<p><strong>Step 1:</strong> Let z = re^(iθ) where r > 0 and θ is the argument of z. Then z̄ = re^(-iθ).</p><p><strong>Step 2:</strong> For a regular n-sided polygon centered at origin with z and z̄ as adjacent vertices, the angle between consecutive vertices is 2π/n. Since z and z̄ are reflections across the real axis, the angle from z to z̄ is 2θ, so: 2θ = 2π/n, which gives θ = π/n.</p><p><strong>Step 3:</strong> Given that Im(z)/Re(z) = tan(θ) = √2 - 1.</p><p><strong>Step 4:</strong> We need to find θ such that tan(θ) = √2 - 1. Note that tan(22.5°) = tan(π/8) = √2 - 1 (this can be verified using the half-angle formula: tan(π/8) = tan(45°/2) = (1 - cos(45°))/sin(45°) = (1 - 1/√2)/(1/√2) = √2 - 1).</p><p><strong>Step 5:</strong> Therefore θ = π/8.</p><p><strong>Step 6:</strong> From Step 2: θ = π/n, so π/8 = π/n, which gives n = 8.</p><p><strong>∴ Answer:</strong> D</p>
Correct Answer: D