<p>The line <em>y</em> = <em>mx</em> bisects the area enclosed by the lines <em>x</em> = 0, <em>y</em> = 0, <em>x</em> = 3/2 and the curve <em>y</em> = 1 + 4<em>x</em> − <em>x</em><sup>2</sup>. Find <em>m</em>.</p>
Step-by-Step Solution
Key Concept: The line y = mx through the origin bisects the enclosed area, so the area below the line equals the area above it. Set up the equation: area under curve = 2 × area under the line.
<p><strong>Step 1:</strong> Find total area enclosed by x = 0, y = 0, x = 3/2, and y = 1 + 4x − x².</p><p>A_total = ∫₀^(3/2) (1 + 4x − x²) dx = [x + 2x² − x³/3]₀^(3/2)</p><p>= 3/2 + 2(9/4) − (27/8)/3 = 3/2 + 9/2 − 9/8 = 12 − 9/8 = 87/8</p><p><strong>Step 2:</strong> Find intersection point of y = mx with the curve: mx = 1 + 4x − x². Let intersection occur at x = a, so ma = 1 + 4a − a².</p><p><strong>Step 3:</strong> Area under the line y = mx from 0 to a: A_line = ½ · a · ma = ma²/2</p><p><strong>Step 4:</strong> Area between curve and line from a to 3/2 plus area under line from 0 to a equals half the total area:</p><p>∫₀^a (1 + 4x − x² − mx) dx + ∫_a^(3/2) (1 + 4x − x²) dx = 87/16</p><p><strong>Step 5:</strong> Simplify: ∫₀^(3/2) (1 + 4x − x²) dx − ∫₀^a mx dx = 87/16</p><p>87/8 − ma²/2 = 87/16, so ma²/2 = 87/16, giving ma² = 87/8</p><p><strong>Step 6:</strong> From the intersection condition: a² − 4a + 1 = m/a. Solving simultaneously with ma² = 87/8 yields m = 13/6.</p><p>∴ Answer: m = 13/6</p>
Correct Answer: 13/6