Let T be the tangent to the parabola y^2 = 16x at the point (64, 32) . Let L be the tangent to the same parabola at another point (x₁, y₁) on the parabola. If L and T are perpendicular to each other, then the distance between the point (x₁, y₁) and the focus of the parabola, is

Step-by-step solution

The equation of the parabola is y^2 = 16x . Comparing with y^2 = 4ax , we get a = 4 . Let the parametric coordinates of the point (64, 32) be (at₁^2, 2at₁) . 2at₁ = 32 8t₁ = 32 t₁ = 4 The slope of the tangent T at t₁ is m₁ = 1 t₁ = 1 4 . Let the tangent L be at the point (x₁, y₁) with parameter t₂ . Its slope is m₂ = 1 t₂ . Since T and L are perpendicular, m₁ m₂ = -1 . 1 4 1 t₂ = -1 t₂ = - 1 4 The x -coordinate of the point (x₁, y₁) is x₁ = at₂^2 = 4 (- 1 4 )^2 = 1 4 . The distance of a point (x₁, y₁) on the parabo