$y=-2x+12a$ is a normal to the parabola $y^2=$

$4ax$ at the point whose distance from the directrix
of the parabola is

Let $P$ be $\left(9t^2/4,9t/2\right),$ then $NP=9t/2.MN=9t/4$

equation of $MQ$ is $y=9t/4$ which meets the parabola

$y^2=9x$ at the point $Q\left(9t^2/16,9t/4\right),N$ is $\left(9t^2/4,0\right)$

equation of $NQ$ is 

$y=-\frac{9\times 16}{4\times 27t}\left(x-\frac{9}{4}{t}^2\right)$ which meets the tangent

$x=0$ (at the vertex $A$) at the point for which

$y=3t$ so that $AT=3t$ and $\frac{AT}{NP}=\frac{3t\times 2}{9t}=\frac{2}{\beta }\text{. }$