<p>The locus of the middle points of chords of the parabola \(y^2 = 4x\), which are of constant length \('2l'\) is:</p>
<p>(a) \((4x + y^2)(y^2 - 4) = 4l^2\)</p>
<p>(b) \((4y + x^2)(x^2 - 4) = 4l^2\)</p>
<p>(c) \((4y - x^2)(x^2 + 4) = 4l^2\)</p>
<p>(d) \((4x - y^2)(y^2 + 4) = 4l^2\)</p>
Step-by-Step Solution
Key Concept: Use parametric form of parabola to express chord endpoints, apply the chord length constraint, and relate the midpoint coordinates to eliminate the parameter, yielding a locus equation.
<p><strong>Step 1: Parametric representation</strong> For the parabola y² = 4x, use parametric form: a point on the parabola is (t², 2t).</p><p><strong>Step 2: Set up two points on a chord</strong> Let the endpoints of a chord be P₁ = (t₁², 2t₁) and P₂ = (t₂², 2t₂).</p><p><strong>Step 3: Apply chord length constraint</strong> The chord length is 2l, so:<br/>(t₁² - t₂²)² + (2t₁ - 2t₂)² = 4l²<br/>Expanding: (t₁ - t₂)²(t₁ + t₂)² + 4(t₁ - t₂)² = 4l²<br/>(t₁ - t₂)²[(t₁ + t₂)² + 4] = 4l²</p><p><strong>Step 4: Find the midpoint</strong> Let M(h, k) be the midpoint of the chord:<br/>h = (t₁² + t₂²)/2<br/>k = (t₁ + t₂)</p><p><strong>Step 5: Express in terms of midpoint coordinates</strong> From Step 4: t₁ + t₂ = k, so k² = (t₁ + t₂)²<br/>Also: t₁² + t₂² = (t₁ + t₂)² - 2t₁t₂ = k² - 2t₁t₂<br/>Thus: h = (k² - 2t₁t₂)/2, which gives t₁t₂ = (k² - 2h)/2</p><p><strong>Step 6: Find (t₁ - t₂)²</strong> (t₁ - t₂)² = (t₁ + t₂)² - 4t₁t₂ = k² - 4·(k² - 2h)/2 = k² - 2k² + 4h = 4h - k²</p><p><strong>Step 7: Substitute into chord length equation</strong> From Step 3: (t₁ - t₂)²[(t₁ + t₂)² + 4] = 4l²<br/>(4h - k²)(k² + 4) = 4l²</p><p><strong>Step 8: Replace h, k with x, y</strong> The locus equation becomes:<br/>(4x - y²)(y² + 4) = 4l²</p><p><strong>∴ Answer:</strong> d</p>
Correct Answer: d