Quadratic Equations
Roots in GP
Grade 11
Question:
<p>Let <strong>α</strong>, <strong>β</strong> be the roots of the equation <strong>x</strong><sup>2</sup> − <strong>x</strong> + <strong>p</strong> = 0 and <strong>γ</strong>, <strong>δ</strong> be the roots of the equation <strong>x</strong><sup>2</sup> − 4<strong>x</strong> + <strong>q</strong> = 0. If α, β, γ and δ are in GP, the integral values of <strong>p</strong> and <strong>q</strong> respectively, are</p>
<p>(a) −2, −32</p>
<p>(b) −2, 3</p>
<p>(c) −6, 3</p>
<p>(d) −6, −32</p>
Step-by-Step Solution
Key Concept: Use the condition that four quantities are in GP to relate the roots of two quadratic equations, then apply Vieta's formulas.
<p><strong>Step 1:</strong> Let <strong>r</strong> be the common ratio of the GP. Then β = αr, γ = αr<sup>2</sup>, and δ = αr<sup>3</sup></p><p><strong>Step 2:</strong> From α + β = 1: α + αr = 1, so α(1 + r) = 1 ... (i)</p><p><strong>Step 3:</strong> From αβ = p: α(αr) = p, so α<sup>2</sup>r = p ... (ii)</p><p><strong>Step 4:</strong> From γ + δ = 4: αr<sup>2</sup> + αr<sup>3</sup> = 4, so αr<sup>2</sup>(1 + r) = 4 ... (iii)</p><p><strong>Step 5:</strong> From (i) and (iii): αr<sup>2</sup> · 1/α = 4, so r<sup>2</sup> = 4, thus r = ±2</p><p><strong>Step 6:</strong> If r = 2: α(1 + 2) = 1 gives α = 1/3, and p = (1/3)<sup>2</sup> · 2 = 2/9 (not integer)</p><p><strong>Step 7:</strong> If r = −2: α(1 − 2) = 1 gives α = −1, and p = (−1)<sup>2</sup>(−2) = −2</p><p><strong>Step 8:</strong> From γδ = q: αr<sup>2</sup> · αr<sup>3</sup> = q, so α<sup>2</sup>r<sup>5</sup> = q. With α = −1 and r = −2: q = 1 · (−32) = −32</p><p>∴ Answer is (a)</p>
Correct Answer: a