Prove that the locus of the middle point of portion of a normal to y2

1.	Prove that the locus of the middle point of portion of a normal to y2 = 4ax intercepted between the curve& the axis is another parabola. Find the vertex & the latus rectum of the second parabola.
Answer» We take a point (at^2,2at) on the parabola the normal is y-2at=-t(x-at^2) y-2at=-xt+at^3 on x axis y=0,(x-2a)t=at^3 x=a(2+t^3) let the locus be (h,k) now find midpoint get he relation between the points.

Prove that the locus of the middle point of portion of a normal to y2 = 4ax intercepted between the curve& the axis is another parabola. Find the vertex & the latus rectum of the second parabola.

Answer»

We take a point (at^2,2at) on the parabola

the normal is

y-2at=-t(x-at^2)

y-2at=-xt+at^3

on x axis y=0,(x-2a)t=at^3

x=a(2+t^3)

let the locus be (h,k)

now find midpoint get he relation between the points.