The slope of the tangent to the curve x=at^2, y=2at at 't' is :

slope of taregent = $\frac{d y}{d x}$

So $x=a t^2, y=2 a t$

so differentiating x, y w.r.t t

so $\frac{d x}{d t}=2 at$         ........(1)

$\frac{d y}{d t}=2 a$         ........(2)

dividing (2) by (1)

we get $\frac{d y}{d x}=\frac{2 a}{2 a t}=\frac{1}{t}$