Question

The chords of the hyperbola x² - y² = a touch the parabola y² = ax. Find the locus of their midpoint.

Answer 1

The equation of the chord of x² - y² = a² whose midpoint is (x₁,y₁) is

xx₁ - yy₁ = x²₁ - y²₁

This chord touches the parabola y² = 4ax  . This implies

Therefore, the locus of (x₁,y₁) is y²(x - a) = x³.