Let chord PQ of length 3 13 of the parabola y^2 = 12x be such that the ordinates of points P and Q are in the ratio 1:2 . If the chord PQ subtends an angle at the focus of the parabola, then is equal to:

Step-by-step solution

Let the coordinates of points P and Q on the parabola y^2 = 12x be (x₁, y₁) and (x₂, y₂) . Given that the ordinates are in the ratio 1:2 , we have y₂ = 2y₁ . Since P and Q lie on the parabola, their abscissae are x₁ = y₁^2 12 and x₂ = y₂^2 12 = 4y₁^2 12 = y₁^2 3 . The length of the chord PQ is 3 13 , so PQ^2 = 117 . Using the distance formula: (x₂ - x₁)^2 + (y₂ - y₁)^2 = 117 ( y₁^2 3 - y₁^2 12 )^2 + (2y₁ - y₁)^2 = 117 ( y₁^2 4 )^2 + y₁^2 = 117 y₁^4 16 + y₁^2 - 117 = 0 y₁^4 + 16y₁^2 - 1872 = 0 (y₁^2 + 52)(y₁^2 - 36)