Applications of Derivatives
Tangents to curves
Grade None
Question:
<p>Angle between the tangents to the curve \(y = x^2 - 5x + 6\) at the points \((2, 0)\) and \((3, 0)\) is</p>
<p>\(\dfrac{\pi}{2}\)</p>
<p>\(\dfrac{\pi}{2}\)</p>
<p>\(\dfrac{\pi}{6}\)</p>
<p>\(\dfrac{\pi}{4}\)</p>
Step-by-Step Solution
Key Concept: Find slopes of tangents at both points using dy/dx, then use the angle formula tan(θ) = |(m₁ - m₂)/(1 + m₁m₂)| to find the angle between them.
<p><strong>Step 1:</strong> Find the derivative to get slope of tangent.</p><p>y = x² - 5x + 6</p><p>dy/dx = 2x - 5</p><p><strong>Step 2:</strong> Find slopes at both points.</p><p>At (2, 0): m₁ = 2(2) - 5 = 4 - 5 = -1</p><p>At (3, 0): m₂ = 2(3) - 5 = 6 - 5 = 1</p><p><strong>Step 3:</strong> Use angle formula between two lines.</p><p>tan(θ) = |m₁ - m₂|/(1 + m₁m₂)</p><p>tan(θ) = |(-1) - (1)|/(1 + (-1)(1))</p><p>tan(θ) = |-2|/(1 - 1) = 2/0 = ∞</p><p><strong>Step 4:</strong> When tan(θ) = ∞, the angle θ = 90°</p><p>The tangents are perpendicular to each other.</p><p>∴ Answer: A (90°)</p>
Correct Answer: A