The angle, between the tangents drawn from the point (1,4) to the parabola y ^2=4 x , is

Step-by-step solution

The point (1,4) serves as the external point from which tangents are drawn to the parabola y^2 = 4x , where a=1 . The equation of a tangent to the parabola with slope m is y = mx + 1 m . Since the tangent passes through (1,4) , substitution yields 4 = m + 1 m , leading to the quadratic m^2 - 4m + 1 = 0 . The slopes m₁ and m₂ are its roots; by Vieta’s formulas, m₁ + m₂ = 4 and m₁ m₂ = 1 . The angle between the tangents is given by = | m₁ - m₂ 1 + m₁ m₂ | . Compute (m₁ - m₂)^2 = (m₁ + m₂)^2 - 4m₁ m₂ = 16 - 4 = 12 , s