Question

The normals at P and Q of the parabola y² = 4ax meet at a point (x₁, y₁) on the curve. Show that

(PQ)² = (x₁ + 4a)(x₁ - 8a)

Answer 1

Let P = (at²₁, 2at₁) and Q = (at₂², 2at₂) Suppose (x₁, y₁) = (at₃²,2at₃). Therefore, we have

Using Eqs. (1) and (2) in Eq. (3), we have

(PQ)² =  (x₁ - 2a - 2a - 4a) (x₁ - 2a - 2a - +  8a)

=(x₁ - 8a)(x₁ + 4a)