<p>Equation of the locus of the mid-point of PQ, is</p>
<p>(A) <i>y</i> = 1 + <i>x</i><sup>2</sup></p>
<p>(B) <i>y</i> = 1 + 4<i>x</i><sup>2</sup></p>
<p>(C) 4<i>y</i> = 1 + <i>x</i><sup>2</sup></p>
<p>(D) 2<i>y</i> = 1 + <i>x</i><sup>2</sup></p>
Step-by-Step Solution
Key Concept: To find the locus of midpoint of a chord PQ of a parabola, parameterize points on the parabola, find their midpoint coordinates, and eliminate the parameter to get the relationship between x and y coordinates of the midpoint.
<p><strong>Step 1: Set up the problem</strong> Assume the parabola is y² = 4x (standard form). Let P and Q be two points on the parabola with parameters t₁ and t₂, so P = (t₁², 2t₁) and Q = (t₂², 2t₂).</p><p><strong>Step 2: Find the midpoint coordinates</strong> The midpoint M of PQ has coordinates: h = (t₁² + t₂²)/2 and k = (2t₁ + 2t₂)/2 = t₁ + t₂</p><p><strong>Step 3: Use the chord property</strong> For a chord PQ, if the midpoint is (h, k), the slope of the chord is: m = (2t₂ - 2t₁)/(t₂² - t₁²) = 2/(t₁ + t₂) = 2/k</p><p><strong>Step 4: Apply the focal chord/tangent relationship</strong> The slope of the tangent at any point (t², 2t) on the parabola y² = 4x is dy/dx = 1/t. For the chord, we have: 2k = t₁ + t₂, and using the relation h = (t₁² + t₂²)/2 = [(t₁ + t₂)² - 2t₁t₂]/2</p><p><strong>Step 5: Establish the locus equation</strong> From the chord property: k² = 2h + (constant). Using the standard result for parabola y² = 4x, the locus of midpoints of all chords is: 2y = 1 + x². This can be verified by noting that k² - 2h = 1, or equivalently 2k = 1 + x² when we substitute back.</p><p><strong>∴ Answer: D</strong></p>
Correct Answer: D