<p>The centre of the circle represented by <i>|z + 1| = 2|z - 1|</i> on the complex plane, is</p>
Step-by-Step Solution
Key Concept: The equation |z + 1| = 2|z - 1| represents the locus of points whose distance from (-1, 0) is twice their distance from (1, 0). This locus forms a circle, and we need to find its center by converting to standard form.
<p><strong>Step 1:</strong> Let z = x + iy where x, y ∈ ℝ. Then |z + 1| = 2|z - 1| becomes:</p><p>|(x+1) + iy| = 2|(x-1) + iy|</p><p><strong>Step 2:</strong> Taking moduli:</p><p>√[(x+1)² + y²] = 2√[(x-1)² + y²]</p><p><strong>Step 3:</strong> Square both sides to eliminate square roots:</p><p>(x+1)² + y² = 4[(x-1)² + y²]</p><p><strong>Step 4:</strong> Expand the left side:</p><p>x² + 2x + 1 + y² = 4[x² - 2x + 1 + y²]</p><p><strong>Step 5:</strong> Expand the right side:</p><p>x² + 2x + 1 + y² = 4x² - 8x + 4 + 4y²</p><p><strong>Step 6:</strong> Rearrange all terms to one side:</p><p>x² + 2x + 1 + y² - 4x² + 8x - 4 - 4y² = 0</p><p>-3x² + 10x - 3 - 3y² = 0</p><p><strong>Step 7:</strong> Divide by -3:</p><p>x² - (10/3)x + 1 + y² = 0</p><p><strong>Step 8:</strong> Complete the square for x:</p><p>x² - (10/3)x + (5/3)² - (5/3)² + 1 + y² = 0</p><p>(x - 5/3)² + y² = (5/3)² - 1 = 25/9 - 9/9 = 16/9</p><p><strong>Step 9:</strong> This is a circle with center at (5/3, 0), so the real part of the center is 5/3.</p><p><strong>∴ Answer:</strong> B</p>
Correct Answer: B