Let $PQ$ be a variable focal chord of the parabola ${y^2} = 4ax\left( {a > 0} \right)$ whose vertex is A. then the locus of centroid of $\Delta APQ$ lies on a parabola whose length of latus rectum is-

Parametric point on ellipse,

$\begin{array}{l} P=\left(a t_{1}^{2}, 2 a t_{1}\right) \\ Q=\left(a t_{2}^{2}, 2 a t_{2}\right) \end{array}$

As $P Q$ passes through focus,

$A_{1} t_{2}=-1$

$\begin{array}{l}\text { Sentroid of } \triangle P A Q=\left(\frac{a t_{1}^{2}+a t_{2}^{2}}{3}, \frac{2 a t_{1}+2 a t_{2}}{3}\right) \\ x=\frac{a}{3}\left(t_{1}^{2}+t_{2}^{2}\right)…(i) \\ y=\frac{2 a}{3}\left(t_{1}+t_{2}\right) \\ \Rightarrow\left(t_{1}+t_{2}\right)=\frac{3 y}{2 a} \\\end{array}$

From (i), $x=\frac{a}{3}\left[\left(t_{1}+t_{2}\right)^{2}-2 t_{1} t_{2}\right]$

$\begin{array}{l} \Rightarrow x=\frac{a}{3}\left[\left(\frac{3 y}{2 a}\right)^{2}-2(-1)\right] \\ \Rightarrow x=\frac{a}{3}\left(\frac{9 y^{2}}{4 a^{2}}+2\right) \\ \Rightarrow 3 x=\frac{9 y^{2}+8 a^{2}}{4 a} \end{array}$

$\begin{array}{l}\Rightarrow 12 a x=9 y^{2}+8 a^{2} \\ \Rightarrow 9 y^{2}=4 a(3 x-2 a) \\ \Rightarrow y^{2}=\frac{4 a}{9} \times 3\left[x-\frac{2 a}{3}\right] \\ \Rightarrow y^{2}=\frac{4 a}{3}\left[x-\frac{2 a}{3}\right]\end{array}$

$\therefore$ length of lotus rectum = $\frac{4 a}{3}$