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

• Correct answer
  A

  vertex is $\left(\frac{2 a}{3}, 0\right)$ it meet its axis at points $T$ and $N$,
• B

  directrix is $x=0$ respectively. The locus of the centroid of $\triangle P T N$ is a parabola whose
• C

  latusrectum is $\frac{2 a}{3}$
• Correct answer
  D

  focus is $(a, 0)$

Solution

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. $(a, 0)$

Hence, the correct answer is options: 'A', 'D'.