Question

Prove that the locus of the midpoint of the portion of a normal to y² = 4ax intercepted between the curve and the axis is another parabola

Answer 1

Normal at P(at², 2at) is

tx + y = 2at + at³

 Substituting y = 0 in the above equation, we have

x = 2a + at²

 Hence, the normal meets the axis at N(2a + at²,0). Let (α    β) be the midpoint of PN so that α = a + at² and β = at. Therefore

Hence, the locus of the midpoint of PN is

y² = a(x - a)

 which is a parabola.