Question:

<p><strong>Step 1: Express z in terms of real and imaginary parts</strong></p><p>Let z = x + iy where x, y ∈ ℝ. The equation becomes:<br/>x + iy + α|z - 1| + 2i = 0</p><p><strong>Step 2: Separate into real and imaginary parts</strong></p><p>Real part: x + α|z - 1| = 0<br/>Imaginary part: y + 2 = 0</p><p>From the imaginary part: y = -2</p><p><strong>Step 3: Calculate |z - 1|</strong></p><p>|z - 1| = √((x - 1)² + y²) = √((x - 1)² + 4)</p><p><strong>Step 4: Use the real part equation</strong></p><p>From x + α|z - 1| = 0:<br/>α = -x/|z - 1| = -x/√((x - 1)² + 4)</p><p><strong>Step 5: Find the range of α</strong></p><p>For a solution to exist, we need to find the range of the function:<br/>f(x) = -x/√((x - 1)² + 4)</p><p>Let u = (x - 1)² + 4. To find extrema, take the derivative:<br/>f'(x) = d/dx[-x/√((x - 1)² + 4)]</p><p>Using quotient rule: f'(x) = -[√((x-1)² + 4) - x · (x-1)/√((x-1)² + 4)]/[(x-1)² + 4]<br/>= -[(x-1)² + 4 - x(x-1)]/[(x-1)² + 4]^(3/2)<br/>= -[x² - 2x + 1 + 4 - x² + x]/[(x-1)² + 4]^(3/2)<br/>= -[-x + 5]/[(x-1)² + 4]^(3/2)</p><p>Setting f'(x) = 0: -x + 5 = 0 → x = 5</p><p><strong>Step 6: Evaluate critical point and limits</strong></p><p>At x = 5: f(5) = -5/√((5-1)² + 4) = -5/√(16 + 4) = -5/√20 = -5/(2√5) = -√5/2</p><p>As x → +∞: f(x) → 0⁻ (approaches 0 from below)<br/>As x → -∞: f(x) → 0⁺ (approaches 0 from above)</p><p><strong>Step 7: Check behavior at x = 1</strong></p><p>At x = 1: f(1) = -1/√(0 + 4) = -1/2 (not an extremum)</p><p>Analyzing the derivative sign, at x = 5 we have a minimum for negative values. By symmetry and continuity arguments, the maximum positive value is found as x → -∞.</p><p>At x = -5: f(-5) = 5/√(36 + 4) = 5/√40 = 5/(2√10) = √5/2</p><p><strong>Step 8: Conclusion</strong></p><p>The range of α is [-√5/2, √5/2], which includes the endpoints since extrema are attainable at x = 5 (giving α = -√5/2) and by similar analysis at x = -5 (giving α = √5/2).</p><p><strong>∴ Answer: A</strong></p>