PQ is double ordnate of the parabola Y squared equal to 4 x the locus of its point of trisection is?

Question

Answer

The equation y^2 = 4ax can be parametrised as:

x = at^2, y = 2at.

If the extremities of the double ordinate are (at^2, 2at) and (at^2, - 2at), then its points of trisection have ordinates y1 and y2 where:

y1 = - 2at + 4at / 3 = 2at / 3

y2 = - 2at + 8at / 3 = - 2at / 3

For both of these points:

y^2 = 4a^2 t^2 / 9

y^2 = (4a / 9)(at^2)

y^2 = 4(a / 9)x.

The locus is the parabola y^2 = 4bx where:

b = a / 9.