Question

In a parabola, the midpoints of parallel chords lie on a line which is a diameter of the parabola.

Answer 1

See Fig. Suppose M(x₁, y₁) be the midpoint of a chord which is parallel to the line lx + my + n = 0. equation of the chord whose midpoint is (x₁, y₁) is

This chord is parallel to the line

lx + my + n = 0

⇒ 2am + ly₁ = 0

This implies that (x₁, y₁) lies on the line 2am + ly = 0, which is horizontal, and hence it is a diameter.