Finding parabola parameter given 2 points

Question

How can I determine which is the directrix and the focus of a parabola and what is the distance between those points, only knowing that this parabola has its symmetry axis = OX and its passes through the points P1 and P2?

EDIT:

Guys, if possible, someone post an example using real numbers please, I think it will be more clear to me (both answers are great, but I'm still having problems to understand, sorry). I'm working with points P1(0,0) P2(6,6) and I need to found p (distance between directrix and focus). Can you guys explain using these numbers?

Answer 1

Let's use the following form for the parabola:

$$y=a(x-h)^2+k$$

Distance from vertex $(h,k)$ to your directrix and focus is calculated using the following formula where $p$ is the distance.

$$a=\frac{1}{4p}$$

If $a$ is positive, the equation for your directrix will be as follows:

$$y=k-p$$

Also, the coordinates of the focus will be the following with $+a$:

$$(h,k+p)$$

If $a$ is negative, there are simply a few sign changes. Directrix equation with $-a$:

$$y=k+p$$

Focus with $-a$:

$$(h, k-p)$$

Hopefully this helped.

Answer 2

The standard equation of the parabola with its axis (symmetric axis=OX) coincident with the x-axis & vertex at the point $(k, 0)$ on the x-axis is $$\color{blue}{y^2=4a(x-x_1)}$$

Where, $a$ & $k$ are arbitrary constants.

Now, satisfying the above equation of parabola by the coordinates of two given point $P_1(x_1, y_1)$ & $P_2(x_2, y_2)$. We get two linear equations in terms of $a$ & $k$ which are determined by solving them. Then we have

The equation of the directrix: $$x-k=-a\iff \color{red}{x=k-a}$$

The focus of the parabola: $$(x-k=a, y=0)\equiv\color{red}{(k+a, 0)}$$