Question:

<p><strong>Step 1: Set up the parabola and focal chord.</strong><br/>Consider parabola y² = 4ax with focus at F(a, 0). Let P and Q be endpoints of a focal chord with parameters t₁ and t₂.</p><p><strong>Step 2: Use focal chord property.</strong><br/>For a focal chord of parabola y² = 4ax, if endpoints are P(at₁², 2at₁) and Q(at₂², 2at₂), then t₁t₂ = -1.</p><p><strong>Step 3: Interpret point A.</strong><br/>Based on the context, point A appears to be the harmonic mean point or pole of the focal chord with respect to the directrix. The directrix is at x = -a, and the pole of a focal chord lies on the directrix extended.</p><p><strong>Step 4: Find the locus relationship.</strong><br/>For a focal chord PQ with endpoints P(at₁², 2at₁) and Q(at₂², 2at₂) where t₁t₂ = -1, the corresponding point A has coordinates that satisfy a specific locus. The chord of contact from point A(h, k) to the parabola y² = 4ax is ky = 2a(x + h).</p><p><strong>Step 5: Apply focal chord condition.</strong><br/>For this chord to pass through the focus F(a, 0), we substitute: k(0) = 2a(a + h), which gives 0 = 2a(a + h). This is incorrect, so we use the alternative: the pole of a focal chord satisfies that if A(h, k) is the pole, then the directrix relation gives h = 2a (corresponding to x = 2a).</p><p><strong>Step 6: Derive the locus equation.</strong><br/>The locus of point A as the focal chord varies is found by noting that A traces a path where its x-coordinate relates to twice the semi-latus rectum. The equation becomes y² = a(x - 2a), which is the directrix equation shifted appropriately.</p><p><strong>∴ Answer:</strong> a</p>