Sequences & Series
Arithmetic Progression
Grade None
Question:
<p>Let <i>α</i> and <i>β</i> be the roots of the equation <i>px</i><sup>2</sup> + <i>qx</i> + <i>r</i> = 0, <i>p</i> ≠ 0. If <i>p</i>, <i>q</i>, <i>r</i> are in AP and \(\dfrac{1}{\alpha} + \dfrac{1}{\beta} = 4\), then the value of |α − β| is</p>
<p>\(\dfrac{\sqrt{34}}{9}\)</p>
<p>\(\dfrac{2\sqrt{13}}{9}\)</p>
<p>\(\dfrac{\sqrt{61}}{9}\)</p>
<p>\(\dfrac{2\sqrt{17}}{9}\)</p>
Step-by-Step Solution
Key Concept: Since p, q, r are in AP, we have 2q = p + r. Combined with Vieta's formulas and the condition 1/α + 1/β = 4, we can express everything in terms of one variable and solve for |α − β| using the discriminant.
<p><strong>Step 1:</strong> From Vieta's formulas: α + β = -q/p and αβ = r/p</p><p><strong>Step 2:</strong> Given 1/α + 1/β = 4, we have (α + β)/(αβ) = 4</p><p>Therefore: (-q/p)/(r/p) = 4 ⟹ -q/r = 4 ⟹ q = -4r</p><p><strong>Step 3:</strong> Since p, q, r are in AP: 2q = p + r</p><p>Substituting q = -4r: 2(-4r) = p + r ⟹ -8r = p + r ⟹ p = -9r</p><p><strong>Step 4:</strong> The quadratic becomes: -9rx² - 4rx + r = 0</p><p>Dividing by -r (r ≠ 0): 9x² + 4x - 1 = 0</p><p><strong>Step 5:</strong> Using |α − β| = √[(α + β)² - 4αβ]</p><p>From 9x² + 4x - 1 = 0: α + β = -4/9 and αβ = -1/9</p><p>|α − β| = √[(-4/9)² - 4(-1/9)] = √[16/81 + 4/9] = √[16/81 + 36/81] = √[52/81] = (2√13)/9</p><p>∴ Answer: D</p>
Correct Answer: D