Equation of Parabola in Reduced Form

Theorem

Let $K$ be a parabola aligned in a cartesian plane in reduced form.

That is:

$(1)$ its focus is at the point $\tuple {c, 0}$

$(2)$ its directrix is aligned with the line $x = -c$

for some $c \in \R_{> 0}$.

The equation of $K$ is:

$y^2 = 4 a x$

The equation of $K$ in parametric form is:

$\ds x$

$=$

$\ds a t^2$

$\ds y$

$=$

$\ds 2 a t$