The tangent \(P T\) and the normal \(P N\) to the parabola \(y^2=4 a x\) at a point \(P\) on

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. \((a, 0)\)