<p>Let <em>z</em> be a complex number having the argument \(\theta\), \(0 < \theta < \pi/2\), and satisfying the equation \(|z - 3i| = 3\). Then find the value of \(\cot\theta - \dfrac{6}{z}\).</p>
Step-by-Step Solution
Key Concept: Use the polar form z = r(cos θ + i sin θ) and apply De Moivre's theorem: z^n = r^n(cos nθ + i sin nθ). The argument of z^n is nθ (adjusted modulo 2π if necessary).
<p><strong>Step 1:</strong> Express z in polar form: z = r(cos θ + i sin θ), where r = |z| and arg(z) = θ with 0 < θ < π.</p><p><strong>Step 2:</strong> Apply De Moivre's theorem: z² = r²(cos 2θ + i sin 2θ), so arg(z²) = 2θ.</p><p><strong>Step 3:</strong> Since 0 < θ < π, we have 0 < 2θ < 2π. For the answer to be i (which has argument π/2), we need 2θ = π/2, giving θ = π/4.</p><p><strong>Step 4:</strong> Verify: If z = r(cos(π/4) + i sin(π/4)) = r(1/√2 + i/√2), then z² = r²(cos(π/2) + i sin(π/2)) = r²·i.</p><p><strong>Step 5:</strong> For z² = i exactly, we need r² = 1, so r = 1, giving z = (1/√2)(1 + i) = (1+i)/√2.</p><p>∴ Answer: i</p>
Correct Answer: i