Question:

<p><strong>Step 1:</strong> From H.P. condition: a, c, b are in H.P. means 1/a, 1/c, 1/b are in A.P.</p><p>Therefore: 2/c = 1/a + 1/b</p><p>Simplifying: 2/c = (a+b)/(ab) → 2ab = c(a+b) ... (i)</p><p><strong>Step 2:</strong> From G.P. condition: a+c, a+b, b+c are in G.P. means (a+b)² = (a+c)(b+c)</p><p>Expanding: a² + 2ab + b² = ab + ac + bc + c²</p><p>Rearranging: a² + ab + b² = ac + bc + c² ... (ii)</p><p><strong>Step 3:</strong> From equation (i): c = 2ab/(a+b)</p><p>Substitute into equation (ii):</p><p>a² + ab + b² = a·(2ab/(a+b)) + b·(2ab/(a+b)) + (2ab/(a+b))²</p><p>a² + ab + b² = 2a²b/(a+b) + 2ab²/(a+b) + 4a²b²/(a+b)²</p><p><strong>Step 4:</strong> Multiply through by (a+b)²:</p><p>(a² + ab + b²)(a+b)² = 2a²b(a+b) + 2ab²(a+b) + 4a²b²</p><p>Let a+b = s: (a² + ab + b²)s² = 2ab(a+b)s + 4a²b² = 2abs² + 4a²b²</p><p>This simplifies to: (a² + ab + b²)s² - 2abs² - 4a²b² = 0</p><p><strong>Step 5:</strong> Testing a=b: (a² + a² + a²)(2a)² - 2a·a·(2a)² - 4a⁴ = 3a²·4a² - 8a⁴ - 4a⁴ = 12a⁴ - 12a⁴ = 0 ✓</p><p>When a = b: c = 2a²/2a = a, so (a+b)/c = 2a/a = 2</p><p>∴ Answer: <strong>B (or 2)</strong></p>