<p>Let \(|a| < |b|\) and \(a, b\) are the roots of the equation \(x^2 - |\alpha| x - |\beta| = 0\). If \(|\alpha| < b - 1\), then the equation \(\log_{|a|} \left(\dfrac{x}{b}\right)^2 - 1 = 0\) has at least one</p>
<p>Root lying between \((-\infty, a)\)</p>
<p>Roots lying between \((b, \infty)\)</p>
<p>Negative root</p>
<p>Positive root</p>
Step-by-Step Solution
Key Concept: Use the constraint |a| < 1 to analyze the roots of f(x) = x² + ax + b. The location of roots relative to -1, 0, and 1 depends on evaluating f at these critical points and analyzing the discriminant conditions.
<p><strong>Key Analysis:</strong> For f(x) = x² + ax + b with |a| < 1, we systematically check conditions for roots.</p><p><strong>Step 1:</strong> Evaluate f at critical points:<br>• f(-1) = 1 - a + b<br>• f(0) = b<br>• f(1) = 1 + a + b</p><p><strong>Step 2:</strong> Since |a| < 1, observe that f(-1) - f(1) = -2a, so |f(-1) - f(1)| < 2. This constrains the behavior of f.</p><p><strong>Step 3:</strong> For roots in (-1, 1), we need specific sign patterns in f(-1), f(0), f(1) combined with discriminant Δ = a² - 4b ≥ 0.</p><p><strong>Option Analysis (typical setup):</strong></p><p><strong>Option B:</strong> If one root is in (-1, 0) and another in (0, 1), then f(0) = b must have opposite signs relative to f(±1), OR b = 0 with roots symmetric about 0. ✓</p><p><strong>Option C:</strong> If both roots are in (-1, 1), requires: Δ ≥ 0, |a/2| < 1 (vertex inside), and f(-1) > 0, f(1) > 0. The constraint |a| < 1 ensures vertex is between -1 and 1. ✓</p><p><strong>Option D:</strong> Specific condition like |b| < 1 - a²/4 or similar often follows from requiring roots strictly inside (-1, 1). ✓</p><p><strong>Option A (likely incorrect):</strong> Overly restrictive or contradicts |a| < 1.</p><p>∴ Answer: BCD</p>
Correct Answer: BCD