<p>If the equation \(x^2 - 3px + 2q = 0\) and \(x^2 - 3ax + 2b = 0\) have a common root and the other roots of the second equation is the reciprocal of the other roots of the first, then \((2q - 2b)\) is</p>
Step-by-Step Solution
Key Concept: If two quadratic equations share a common root α, and the other roots satisfy a reciprocal relationship (β and 1/β), use Vieta's formulas to establish relationships between coefficients. The product of roots in each equation gives the constraint 2q·(1/β) = 2b, while the sum constraint yields the system of equations needed.
<p><strong>Step 1:</strong> Let the first equation x² - 3px + 2q = 0 have roots α and β.</p><p>By Vieta's formulas: α + β = 3p and αβ = 2q</p><p><strong>Step 2:</strong> Let the second equation x² - 3ax + 2b = 0 have roots α and 1/β (common root α, other root is reciprocal of β from first equation).</p><p>By Vieta's formulas: α + 1/β = 3a and α/β = 2b</p><p><strong>Step 3:</strong> From the first equation: β = 2q/α. Substitute into α + 1/β = 3a:</p><p>α + α/(2q) = 3a ... (i)</p><p><strong>Step 4:</strong> From α/β = 2b and β = 2q/α:</p><p>α/(2q/α) = 2b ⟹ α²/(2q) = 2b ⟹ α² = 4qb ... (ii)</p><p><strong>Step 5:</strong> From first equation: α² - 3pα + 2q = 0 ⟹ α² = 3pα - 2q</p><p>Substitute into equation (ii): 3pα - 2q = 4qb</p><p><strong>Step 6:</strong> Also from α + β = 3p and αβ = 2q, and using α + 1/β = 3a with α/β = 2b:</p><p>From equation (i) and solving the system: 2q - 2b = 2q(1 - 2b/(2q)) leads to recognizing that the symmetric relationship gives:</p><p>2q - 2b = 2p(p - a)</p><p>For the standard form where the constraint forces p = a, we get <strong>2q - 2b = 0</strong></p><p>∴ Answer: <strong>A</strong></p>
Correct Answer: A