<p>If \(\sin\theta\) and \(\cos\theta\) are the roots of the quadratic equation \(ax^2 + bx + c = 0\) (\(ac \neq 0\)). Then find the value of \(\frac{b^2 - a^2}{ac}\).</p>
Step-by-Step Solution
Key Concept: If sin θ and cos θ are roots of a quadratic equation, use Vieta's formulas to find their sum and product, then apply the fundamental trigonometric identity sin²θ + cos²θ = 1 to establish a relationship between coefficients.
<p><strong>Step 1:</strong> Apply Vieta's formulas. Since sin θ and cos θ are roots of ax² + bx + c = 0:</p><p>Sum of roots: sin θ + cos θ = -b/a</p><p>Product of roots: sin θ · cos θ = c/a</p><p><strong>Step 2:</strong> Square the sum of roots equation:</p><p>(sin θ + cos θ)² = (-b/a)²</p><p>sin²θ + 2sin θ cos θ + cos²θ = b²/a²</p><p><strong>Step 3:</strong> Use the fundamental trigonometric identity sin²θ + cos²θ = 1:</p><p>1 + 2sin θ cos θ = b²/a²</p><p><strong>Step 4:</strong> Substitute sin θ cos θ = c/a from Step 1:</p><p>1 + 2(c/a) = b²/a²</p><p><strong>Step 5:</strong> Multiply both sides by a²:</p><p>a² + 2ac = b²</p><p><strong>Step 6:</strong> Rearrange to find b² - a²:</p><p>b² - a² = 2ac</p><p><strong>Step 7:</strong> Divide both sides by ac (valid since ac ≠ 0):</p><p>(b² - a²)/ac = 2ac/ac = 2</p><p><strong>∴ Answer: 2</strong></p>
Correct Answer: 2