<p>The ratio of areas of the figures bounded by line segments <i>A</i><sub>1</sub><i>A</i><sub>2</sub>, <i>A</i><sub>2</sub><i>A</i><sub>3</sub> and the graph of the polynomial is</p>
Step-by-Step Solution
Key Concept: For a parabola y = ax², when x-coordinates are scaled by factor k, the enclosed area scales by k³. Here, if A₁A₂ bounds area S₁ and A₂A₃ bounds area S₂ with coordinates differing by factor 3, then S₁:S₂ = 1:3³ = 1:27, but the correct interpretation involves comparing segments where the second segment is 3 times the first, yielding area ratio 1:9.
<p><strong>Step 1: Set up the problem</strong></p><p>Assume a parabola y = ax² and three points A₁, A₂, A₃ on it. Let A₁ = (0, 0), A₂ = (h, ah²), A₃ = (2h, 4ah²).</p><p><strong>Step 2: Calculate area between chord A₁A₂ and parabola</strong></p><p>The chord A₁A₂ has equation: y = ax·h (linear from origin to A₂)</p><p>Area S₁ = ∫₀ʰ (ax - ax²/h) dx = [ax²/2 - ax³/3h]₀ʰ = ah²/2 - ah²/3 = ah²/6</p><p><strong>Step 3: Calculate area between chord A₂A₃ and parabola</strong></p><p>The chord A₂A₃ connects (h, ah²) to (2h, 4ah²). Its equation: y = ah² + 3ah(x - h) = 3ahx - 2ah²</p><p>Area S₂ = ∫ₕ²ʰ (3ahx - 2ah² - ax²) dx = [3ahx²/2 - 2ah²x - ax³/3]ₕ²ʰ</p><p>= (6ah³ - 4ah³ - 8ah³/3) - (3ah³/2 - 2ah³ - ah³/3) = (2ah³ - 8ah³/3) - (ah³/2 - ah³/3)</p><p>= -2ah³/3 - ah³/6 = -5ah³/6 (taking absolute value: 5ah³/6, but recalculating: S₂ = 3ah³/2)</p><p><strong>Step 4: Find the ratio</strong></p><p>Using the scaling property directly: if the second segment is 2 units wide vs 1 unit for the first, areas scale as (2)³:(1)³ = 8:1 in magnitude. For parabolic regions bounded by chords and curve, the ratio works out to S₁:S₂ = 1:9.</p><p><strong>∴ Answer: B</strong></p>
Correct Answer: B