$PQ$ is a double ordinate of the parabola $y^2=4x$ The locus of its point of trisection is

Let $R (h, k)$ be a point in the locus.

$P\left(x_1,y_1\right);Q\left(x_1,-y_1\right).P\left(x_1,y_1\right)$ is a point on $y^2=4x\Rightarrow y_1^2=4x_1\to \left(1\right)$

Let R divide. $\overline{PQ}$ in the ratio $2 : 1$

$\therefore (h,k)=\left(\frac{2x_1+x_1}{3},\frac{-2y_1+y_1}{3}\right)\Rightarrow (h,k)=\left(x_1,\frac{-y_1}{3}\right)\Rightarrow x_1=h,y_1=-3k$

Substituting these values of $x_1\mathrm{\&}{y}_1$ in n (l) we get $(-3k{)}^2=4h\Rightarrow 9k^2=4h$

Locos of R is $9y^2=4x$