The equation for a parabola in vertex form is
$\displaystyle y=a(x-h)^2+k$
whereas in older or more advanced references to conics, the formula is
$\displaystyle 4p(y-k)=(x-h)^2$
now immediately the correlation is obvious:
$\displaystyle y=a(x-h)^2+k$
$\displaystyle \frac{1}{a}(y-k)=(x-h)^2$
hence, $\displaystyle 4p = \frac{1}{a}$
the question is simply why? How does $\displaystyle \frac{1}{a}$ become $4p$? In addition, I'm aware that if the directrix is at $y=-p$ and the focus is at $(0,p)$, then the perpendicular distance from the directrix to the focus is $2p$, so obviously $4p$ is therefore twice the distance from the directrix to the focus, but still, why is $4p$ used instead of just $2p$?
$\endgroup$ 41 Answer
$\begingroup$So your question is why do we use the formula
$$ 4p(y-k)=(x-h)^2 $$
rather than the formula
$$ 2p(y-k)=(x-h)^2 $$
Certainly, this could be done, but it would replace each $p$ in the following parabolic relationship diagram with the fraction $\dfrac{p}{2}$. This is the only reason I can think of.
Note also that $4p$ also happens to be the length of the focal chord (Latin: latus rectum).
