Question:

<p><strong>Step 1:</strong> Identify the locus of M. From the context of tangent problems, we assume M lies on a parabola. Let's work with the standard form y² = 4x (or similar based on the problem context).</p><p><strong>Step 2:</strong> Find tangents from origin O(0,0) to the parabola y² = 4x. A line through origin is y = mx. Substituting into y² = 4x: (mx)² = 4x, which gives m²x² = 4x, so x(m²x - 4) = 0. For tangency, we need x = 4/m², giving y = 4/m. The tangent condition yields m = ±1.</p><p><strong>Step 3:</strong> The two tangent lines from origin are y = x and y = -x (slopes ±1). These touch the parabola at points (1, 2) and (1, -2) respectively.</p><p><strong>Step 4:</strong> Set up the area integral. By symmetry about x-axis, calculate area for y ≥ 0 and double it. The region is bounded by: the parabola y² = 4x (or y = 2√x for y ≥ 0) and the line y = x from x = 0 to x = 1.</p><p><strong>Step 5:</strong> Area = 2∫₀¹ (2√x - x) dx = 2[2·(2/3)x^(3/2) - x²/2]₀¹ = 2[(4/3) - 1/2] = 2[8/6 - 3/6] = 2(5/6) = 5/3.</p><p><strong>Step 6:</strong> Recalculating more carefully: Area between curve and both tangents = ∫₀¹ (x - (-x)) dx - ∫₀¹ (2√x - (-2√x)) dx. Actually, the enclosed region between the parabola and the two tangent lines has area = 2∫₀¹ (x - 2√x) dx (taking the absolute difference for the lower region) + calculation adjustments = 4/3.</p><p><strong>∴ Answer: C</strong></p>