In most books I find on conic sections, almost all of the parabolas are found after parameterizing the parabola $y^2=4ax$ as: $y=2at$,$x=at^2$. in problem book I am using, it goes so far to say this is the best parameterization. Of course, best is subjective, but this certainly seems a very 'esteemed' parameterization.
My question: How would one motivate this parameterization? / What is the reason this parameterization gives so many nice properties? Eg:
If the two parameter value correspond to end of focal chord, then: $$t_1t_2=-1$$
The slope of the normal at parameter $t$ is given as $$-t$$ etc