<p>Substituting <em>z = x + iy</em>, the equation <em>2|z + 3i| − |z − i| = 0</em> represents a circle. Find the radius of this circle.</p>
Step-by-Step Solution
Key Concept: Convert the modulus equation into standard form by substituting z = x + iy and squaring strategically to eliminate absolute values, then rearrange into the circle equation (x - h)² + (y - k)² = r².
<p><strong>Step 1:</strong> Substitute z = x + iy into 2|z + 3i| − |z − i| = 0</p><p>2|x + i(y + 3)| = |x + i(y − 1)|</p><p><strong>Step 2:</strong> Expand the moduli using |a + ib| = √(a² + b²)</p><p>2√(x² + (y + 3)²) = √(x² + (y − 1)²)</p><p><strong>Step 3:</strong> Square both sides to eliminate outer radicals</p><p>4[x² + (y + 3)²] = x² + (y − 1)²</p><p>4x² + 4(y² + 6y + 9) = x² + y² − 2y + 1</p><p><strong>Step 4:</strong> Simplify and collect terms</p><p>4x² + 4y² + 24y + 36 = x² + y² − 2y + 1</p><p>3x² + 3y² + 26y + 35 = 0</p><p>x² + y² + (26/3)y + 35/3 = 0</p><p><strong>Step 5:</strong> Complete the square for y-terms</p><p>x² + (y + 13/3)² − 169/9 + 35/3 = 0</p><p>x² + (y + 13/3)² = 169/9 − 105/9 = 64/9</p><p><strong>Step 6:</strong> Identify radius from standard form x² + (y − k)² = r²</p><p>∴ Radius = √(64/9) = <strong>8/3</strong></p>
Correct Answer: A