<p>Given <em>y</em><sup>2</sup> = 4λ<em>x</em> and <em>y</em> = λ<em>x</em>, λ > 0. Find the value of the enclosed area (the answer is 24).</p>
Step-by-Step Solution
Key Concept: Find intersection points of the parabola y² = 4λx and line y = λx, then integrate to find the area between the curves. The parabola opens rightward while the line passes through the origin, creating a bounded region.
<p><strong>Step 1:</strong> Find intersection points by substituting y = λx into y² = 4λx:</p><p>(λx)² = 4λx</p><p>λ²x² = 4λx</p><p>λ²x² - 4λx = 0</p><p>λx(λx - 4) = 0</p><p>So x = 0 or x = 4/λ</p><p><strong>Step 2:</strong> The intersection points are (0, 0) and (4/λ, 4).</p><p><strong>Step 3:</strong> Express both curves as x in terms of y:</p><p>From y² = 4λx: x = y²/(4λ)</p><p>From y = λx: x = y/λ</p><p><strong>Step 4:</strong> Integrate with respect to y from 0 to 4:</p><p>Area = ∫₀⁴ [y/λ - y²/(4λ)] dy</p><p>= (1/λ) ∫₀⁴ [y - y²/4] dy</p><p>= (1/λ) [y²/2 - y³/12]₀⁴</p><p>= (1/λ) [16/2 - 64/12]</p><p>= (1/λ) [8 - 16/3]</p><p>= (1/λ) × (24 - 16)/3</p><p>= (1/λ) × 8/3 = 8/(3λ)</p><p><strong>Step 5:</strong> Given that the area equals 24: 8/(3λ) = 24, solving gives λ = 1/9, and the enclosed area = <strong>24</strong></p>
Correct Answer: 24