Question

Let p and q be non-zero real numbers and x₁, x₂ and x₃ be non-zero real roots of the equation x₃ − 3px₂ - 3qx₂ − 1 = 0. Then show that the centroid of the triangle whose vertices are [x₁, (1/x₁)], [x_2, (1/x₂)] and [x₃, (1/x₃)] is (p, q).

Answer 1

Let y₁ = 1/x₁, y₂ = 1/x₂ and y₃ = 1/x₃. By hypothesis,

x₁ + x₂ + x₃ = 3p

x₁x₂ + x₂x_{3 +} x₃x₁ = 3q

x₁x₂x₃ = 1