<p>Let \(z\) be a complex number satisfying \(|z| = 3|z - 1|\). Then prove that \(\left|z - \dfrac{9}{8}\right| = \dfrac{3}{8}\).</p>
Step-by-Step Solution
Key Concept: The condition |z| = 3|z - 1| represents a circle (Apollonius circle). Convert this geometric constraint into an algebraic equation by squaring and substituting z = x + iy to find the locus, then show this circle has center 9/8 and radius 3/8.
<p><strong>Step 1:</strong> Start with the given condition |z| = 3|z - 1|. Square both sides:</p><p>|z|² = 9|z - 1|²</p><p><strong>Step 2:</strong> Let z = x + iy. Substitute:</p><p>x² + y² = 9[(x - 1)² + y²]</p><p>x² + y² = 9[x² - 2x + 1 + y²]</p><p>x² + y² = 9x² - 18x + 9 + 9y²</p><p><strong>Step 3:</strong> Rearrange terms:</p><p>x² + y² - 9x² + 18x - 9 - 9y² = 0</p><p>-8x² - 8y² + 18x - 9 = 0</p><p>8x² + 8y² - 18x + 9 = 0</p><p><strong>Step 4:</strong> Divide by 8:</p><p>x² + y² - (9/4)x + 9/8 = 0</p><p><strong>Step 5:</strong> Complete the square for x:</p><p>[x² - (9/4)x + (9/8)²] + y² = (9/8)² - 9/8</p><p>[x - 9/8]² + y² = 81/64 - 72/64 = 9/64</p><p><strong>Step 6:</strong> Recognize this as a circle with center (9/8, 0) and radius √(9/64) = 3/8:</p><p>|z - 9/8| = 3/8</p><p>∴ Proof Complete</p>
Correct Answer: Proof