We have, $y=2 a t$
$\frac{d y}{d t}=2 a \frac{d}{d t}(t)=2 a(1)=2 a \\
\text { also } x=a t^{2} \\
\frac{d x}{d x}=a \frac{d}{d t}\left(t^{2}\right)=a(2 t)=2 a t \\
\text { now } \frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{2 a}{2 a t}=\frac{1}{t}$