If PQ is double ordinate of the parabola ${\mathrm{y}}^2=4\mathrm{ax}$ then locus of its point of trisection is

$Given parabola y^2=4ax$

$$\begin{gathered} \mathrm{Let} \mathrm{P}=(a{t}^2,2at),Q=(a{t}^2,-2at) be end points of double ordinate of parabola \\ Let R=(x_{ },y_{ }) be locus of point of trisection of PQ \\ Then R=(\frac{a{t}^2+2a{t}^2}{3},\frac{-2at+4at}{3}) \\ \Rightarrow x_{ }=a{t}^2\to \left(1\right) \\ and y_{ }=\frac{2at}{3}\Rightarrow t=\frac{3y}{2a} \\ from \left(1\right) x=a{\left(\frac{3y}{2a}\right)}^2 \Rightarrow x=\frac{9y^2}{4a} \\ \Rightarrow 9y^2=4ax \end{gathered}$$