<p>Let AB be a line segment of length 4 with \(A\) on the line \(y = 2x\) and \(B\) on the line \(y = x\). The locus of the middle point of the line segment is</p>
Step-by-Step Solution
Key Concept: Parameterize points A and B on their respective lines, use the constraint |AB| = 4, then find the locus of the midpoint M by eliminating the parameter.
<p><strong>Step 1:</strong> Let A = (a, 2a) on line y = 2x and B = (b, b) on line y = x.</p><p><strong>Step 2:</strong> Apply distance constraint: (b - a)² + (b - 2a)² = 16</p><p>Expanding: (b - a)² + (b - 2a)² = b² - 2ab + a² + b² - 4ab + 4a² = 2b² - 6ab + 5a² = 16</p><p><strong>Step 3:</strong> Midpoint M = ((a + b)/2, (2a + b)/2). Let M = (h, k).</p><p>Then: a + b = 2h and 2a + b = 2k</p><p>Solving: a = 2k - 2h and b = 4h - 2k</p><p><strong>Step 4:</strong> Substitute into distance equation:</p><p>2(4h - 2k)² - 6(2k - 2h)(4h - 2k) + 5(2k - 2h)² = 16</p><p>2(16h² - 16hk + 4k²) - 6(8hk - 4k² - 8h² + 4hk) + 5(4k² - 8hk + 4h²) = 16</p><p>32h² - 32hk + 8k² - 48hk + 24k² + 48h² - 20k² + 40hk - 20h² = 16</p><p>60h² - 40hk + 12k² = 16</p><p>15h² - 10hk + 3k² = 4</p><p><strong>Step 5:</strong> Replacing (h, k) with (x, y): 15x² - 10xy + 3y² = 4 (or equivalent form)</p><p>∴ Answer: C</p>
Correct Answer: C