Standard form of parametric equation of Parabola

Question

Standard parametric equations of a parabola of the form $y^2=4ax$ are: $$ x(t)=at^2\\ y(t)=2at $$ which is fine since it can be easily verified. But is there any reason or advantage of making such a choice in the parametric equation of parabola ?

Answer 1

The standard Cartesian form equation for the parabola $y^2=4ax$ is significant because $a$ is the focal length, the focus of the parabola is $(a,0)$ and also because $4a$ is the length of the latus rectum.

For this parabola, the standard parametric equation $(at^2, 2at)$ is probably the simplest possible as it does not contain fractions. Other possibilities are $\left(\frac {t^2}{4a} , t\right), \left(\frac {t^2}a, 2t\right)$, which are not as neat.

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Another example of a possible parametric equation is $\big(4a\sin t, 2a(1-\cos 2t)\big)$.

Answer 2

There is no standard parametrization.

A parametrization is beneficial if the parameter has an extra or particular geometrical or physical significance.

The given parametrization has focal length $a$. Differentiating $x$ wrt $y$ through $t,$ it can be appreciated that $t$ also represents tangent of angle which the tangent of parabola ( axis on $x$ axis) makes to the $y$ axis. It is also simple, algebraically.

EDIT1:

Another direct (unparametrized) oblique axes form with two branches with constants $ {(m,h,k)} $ is:

$$y= m x \pm \sqrt{m x h + k^2}$$