Why does the formula of a parabola in vertex form include $4p$?

Question

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 $2p$?

Answer 1

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).