Remove the original intersection before using a discriminant

A curve is given by $x=at^2$, $y=2at$.

Show that the normal at $(at^2,2at)$ is $y+tx=at^3+2at$.

If that normal meets the curve again at $(aq^2,2aq)$, show that $t^2+qt+2=0$. Deduce that $q^2$ cannot be less than $8$.