The tangent P T and the normal P N to the parabola y^2=4 a x at a...

Equation of tangent and normal at point $P\left(a t^2, 2 a t\right)$ is $t y=x=a t^2$ and $y=-t x+2 a t+a t^2$ Let centroid of $\triangle P T N$ is $R(h, k)$. $\therefore \quad h=\frac{a t^2+\left(-a t^2\right)+2 a+a t^2}{3}$ and $k=\frac{2 a t}{3}$
\begin{array}{rlrl} & \Rightarrow & 3 h & =2 a+a \cdot\left(\frac{3 k}{2 a}\right)^2 \\ \Rightarrow & & 3 h & =2 a+\frac{9 k^2}{4 a} \\ \Rightarrow & 9 k^2 & =4 a(3 h-2 a) \end{array} $\therefore$ Locus of centroid is $$y^2=\frac{4 a}{3}\left(x-\frac{2 a}{3}\right)$$ $\therefore \operatorname{Vertex}\left(\frac{2 a}{3}, 0\right)$; directrix $$x-\frac{2 a}{3}=-\frac{a}{3} \Rightarrow x=\frac{a}{3}$$ Latusrectum $=\frac{4 a}{3}$ $\therefore$ Focus \left(\frac{a}{3}+\frac{2 a}{3}, 0\right)$i.e.$(a, 0)