<p>If \(\left|\dfrac{z-2}{z-3}\right| = 2\) represents a circle, then find its centre and radius.</p>
Step-by-Step Solution
Key Concept: The equation |z-a|/|z-b| = k represents a circle (Apollonius circle). Square both sides to eliminate the modulus, then rearrange into standard form (x-h)² + (y-k)² = r² by substituting z = x + iy.
<p><strong>Step 1:</strong> Start with |z-2|/(z-3)| = 2. Square both sides:</p><p>|(z-2)|²/|(z-3)|² = 4</p><p>|z-2|² = 4|z-3|²</p><p><strong>Step 2:</strong> Substitute z = x + iy:</p><p>|(x-2) + iy|² = 4|(x-3) + iy|²</p><p>(x-2)² + y² = 4[(x-3)² + y²]</p><p><strong>Step 3:</strong> Expand both sides:</p><p>x² - 4x + 4 + y² = 4[x² - 6x + 9 + y²]</p><p>x² - 4x + 4 + y² = 4x² - 24x + 36 + 4y²</p><p><strong>Step 4:</strong> Rearrange to standard form:</p><p>0 = 3x² - 20x + 3y² + 32</p><p>3x² - 20x + 3y² = -32</p><p>x² - (20/3)x + y² = -32/3</p><p><strong>Step 5:</strong> Complete the square for x:</p><p>[x² - (20/3)x + (10/3)²] + y² = -32/3 + 100/9</p><p>[x - (10/3)]² + y² = -96/9 + 100/9 = 4/9</p><p><strong>Step 6:</strong> This is in the form (x-h)² + (y-k)² = r²</p><p>∴ <strong>Centre = (10/3, 0); Radius = 2/3</strong></p>
Correct Answer: Centre = (10/3, 0); Radius = 2/3