The tangent $PT$ and the normal $PN$ to the parabola $y^{2} = 4ax$ at a point $P$ on it meet its axis at points $T$ and $N$, respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose (1) vertex is $\left(\dfrac{2a}{3}, 0\right)$ (2) directrix is $x = 0$ (3) latus rectum is $\dfrac{2a}{3}$ (4) focus is $(a, 0)$

$\text{Parabola: } y^2 = 4ax$

$P \text{ on the parabola; tangent } PT, \text{ normal } PN$

$PT \text{ and } PN \text{ meet the axis at } T \text{ and } N$

To find: locus of the centroid of triangle PTN

$\text{and identify its vertex / directrix / latus rectum / focus}$

$P = (at^2,\ 2at) \quad (t \text{ the parameter})$

$ty = x + at^2 \ \Rightarrow\ T = (-at^2,\ 0)$

$y = -tx + at^3 + 2at \ \Rightarrow\ N = (at^2 + 2a,\ 0)$

$G = \left(\dfrac{x_P+x_T+x_N}{3},\ \dfrac{y_P+y_T+y_N}{3}\right)$

$G_x = \dfrac{at^2 - at^2 + (at^2 + 2a)}{3} = \dfrac{at^2 + 2a}{3}$

$G_y = \dfrac{2at + 0 + 0}{3} = \dfrac{2at}{3}$

$h = \dfrac{at^2 + 2a}{3}, \quad k = \dfrac{2at}{3}$

$t = \dfrac{3k}{2a}$

$h = \dfrac{a}{3}\left(\dfrac{9k^2}{4a^2}\right) + \dfrac{2a}{3}$

$h = \dfrac{3k^2}{4a} + \dfrac{2a}{3}$

$k^2 = \dfrac{4a}{3}\left(h - \dfrac{2a}{3}\right)$

$\text{Vertex } \left(\dfrac{2a}{3},\ 0\right)$

$k^2 = 4a'\left(h - \dfrac{2a}{3}\right), \quad a' = \dfrac{a}{3}$

$\text{Focus} = \left(\dfrac{2a}{3} + \dfrac{a}{3},\ 0\right) = (a,\ 0)$

$\text{Directrix: } x = \dfrac{2a}{3} - \dfrac{a}{3} = \dfrac{a}{3}$

$\text{Latus rectum} = 4a' = \dfrac{4a}{3}$

$(1)\ \checkmark \qquad (2)\ \times \qquad (3)\ \times \qquad (4)\ \checkmark$

$\boxed{y^2 = \dfrac{4a}{3}\left(x - \dfrac{2a}{3}\right)}$

$\boxed{\text{Correct: } (1)\ \text{vertex } \left(\dfrac{2a}{3},0\right),\ (4)\ \text{focus } (a,0)}$