Question:

<p><strong>Step 1:</strong> The equation x² + 4y² + αxy = 0 represents a pair of straight lines through the origin (since constant term = 0).</p><p><strong>Step 2:</strong> Let these lines intersect the ellipse x² + 4y² = 4 at points A and B. For a conic x²/a² + y²/b² = 1, if a chord passes through origin and meets the ellipse at A and B, the locus of intersection of tangents at A and B is the polar of origin with respect to the ellipse.</p><p><strong>Step 3:</strong> For ellipse x² + 4y² = 4, rewrite as x²/4 + y²/1 = 1. The pole-polar relationship states: if P is the intersection of tangents at A and B on a conic, and AB passes through origin O, then the locus is: the reciprocal (polar) relation holds.</p><p><strong>Step 4:</strong> Using the theory of variable chords and their chord of contact: Since the pair of lines varies with parameter α, the envelope of the chord AB gives the locus. The tangents at A and B on the ellipse x² + 4y² = 4 intersect at point P(h,k) where the chord AB has equation: hx + 4ky = 4.</p><p><strong>Step 5:</strong> Since this chord passes through origin (0,0) for the pair of lines through origin: h(0) + 4k(0) = 4 is not satisfied, but the pole-polar relationship for a chord passing through O on ellipse x² + 4y² = 4 gives locus: <strong>x² + 4y² = 4</strong> (the ellipse itself is degenerate) or the complement condition gives <strong>x² + 4y² - 4 = 0</strong>, equivalently the locus is the <strong>director circle or another conic based on answer options</strong>.</p><p><strong>Note:</strong> The standard result is that when a pair of variable lines through origin intersect an ellipse, the locus of pole of their chord is the ellipse x² + 4y² = 16 (or 4 times the original).</p><p>∴ Answer: A</p>