Quadratic Equations
Roots and sequences
Grade 11
Question:
<p>If <em>α</em>, <em>β</em> are the nonzero roots of \(ax^2 + bx + c = 0\) and \(\alpha^2\), \(\beta^2\) are the roots of \(a^2x^2 + b^2x + c^2 = 0\), then <em>a</em>, <em>b</em>, <em>c</em> are in</p>
<p>G.P.</p>
<p>H.P.</p>
<p>A.P.</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas on both equations and relate the sum and product of roots. The condition that α² and β² are roots of the second equation creates a constraint: (α + β)² - 2αβ must equal the ratio of coefficients in the second equation.
<p><strong>Step 1:</strong> For equation ax² + bx + c = 0 with roots α, β:</p><p>α + β = -b/a and αβ = c/a</p><p><strong>Step 2:</strong> For equation a²x² + b²x + c² = 0 with roots α², β²:</p><p>α² + β² = -b²/a² and α²β² = c²/a²</p><p><strong>Step 3:</strong> From α²β² = c²/a²: (αβ)² = c²/a² ⟹ (c/a)² = c²/a² ✓ (consistent)</p><p><strong>Step 4:</strong> From α² + β² = -b²/a²:</p><p>(α + β)² - 2αβ = -b²/a²</p><p>(-b/a)² - 2(c/a) = -b²/a²</p><p>b²/a² - 2c/a = -b²/a²</p><p>2b²/a² = 2c/a</p><p>b²/a = c</p><p><strong>Step 5:</strong> Therefore: b² = ac, which means a, b, c are in <strong>Geometric Progression</strong></p><p>∴ Answer: A (a, b, c are in GP)</p>
Correct Answer: A