<p>Since \(R\) divides \(PQ\) in the ratio \(2:1\), where \(P = (-t^2, 2t)\), \(t > 0\) is on \(y^2 = -4x\) and \(Q = (-t^2, -2t)\). Find the locus of \(R = (x, y)\).</p>
Step-by-Step Solution
Key Concept: Use the section formula to find coordinates of R that divides PQ in ratio 2:1, then eliminate the parameter t to get the locus equation.
<p><strong>Step 1:</strong> Identify points P = (-t², 2t) and Q = (-t², -2t) on parabola y² = -4x.</p><p><strong>Step 2:</strong> Apply section formula. If R divides PQ in ratio 2:1, then:</p><p>R = ((2·(-t²) + 1·(-t²))/(2+1), (2·(-2t) + 1·(2t))/(2+1))</p><p>R = ((-2t² - t²)/3, (-4t + 2t)/3) = (-3t²/3, -2t/3) = (-t², -2t/3)</p><p><strong>Step 3:</strong> From R = (x, y), we have x = -t² and y = -2t/3.</p><p><strong>Step 4:</strong> From y = -2t/3, we get t = -3y/2, so t² = 9y²/4.</p><p><strong>Step 5:</strong> Substitute into x = -t²: x = -9y²/4, which gives y² = -4x/9.</p><p>∴ <strong>Answer: y² = -4x/9 (or 9y² = -4x)</strong></p>
Correct Answer: D