Question Detail

This question requires finding the general form of the tangent line equation and then identifying which of the options matches this form at a point on the curve.First, implicitly differentiate the curve equation y² = 3x² + 1 with respect to x:2y (dy/dx) = 6x => dy/dx = 3x/y.The slope of the tangent at a point (x₀, y₀) is m_t = 3x₀/y₀.The equation of the tangent line at (x₀, y₀) is y - y₀ = m_t (x - x₀). Substituting and simplifying gives 3x₀x - y₀y + 1 = 0.Comparing this general form with Option D, 3x - 2y + 1 = 0, implies x₀ = 1 and y₀ = 2. We verify that the point (1, 2) lies on the curve: 2² = 3(1)² + 1 => 4 = 3 + 1, which is true. Therefore, Option D is the equation of the tangent at (1, 2).