Question

Show that the locus of the midpoints of chords of a parabola passing through the vertex is in turn a parabola whose latus rectum is half of the latus rectum of the original. 

Answer 1

The equation of the chord in terms of its midpoint M(x₁, y₁) is

S₁ = S₁₁

⇒ yy₁ - 2a(x + x₁) = y₁² - 4ax₁

yy₁ - 2ax = y₁² - 2ax₁ which passes through (0,0)

y₁² - 2ax₁ = 0

Hence, the locus of M(x₁, y₁) is the parabola y² = 2ax.