$PN$ is an ordinate of the parabola $y^2=9x$ .$A$ straight
line is drawn through the mid-point $M$of $PN$parallel
to the axis of the parabola meeting the parabola
at $Q$. $NQ$meets the tangent at the vertex $A$, at a
point $T$, then $AT/NP =$

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

equation of $MQ\text{ is }y=9t/4$ which meets the parabola

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

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 $AT=3t_1$and $\frac{AT}{NP}=\frac{3t\times 2}{9t}=\frac{2}{3}\text{. }$