How to derive the equation of a parabola from the directrix and focus Could someone please offer me proof and explanation of the following? - I am just having trouble with finding the '$a$' part of the equation.
"The leading coefficient '$a$' in the equation
$$y−y_1 =​​ a(x−x_1)^{2}$$​
indicates how "wide" and in what direction the parabola opens. It's always the reciprocal of $2$ times the distance from the directrix to the focus."
Thanks.

The parabola is defined by the locus of the points that are at equal distance from the focus and a line, called the directrice. 
Let's call $d$ the distance between the focus $F$ and the directrix $D$. 
Then $a=\dfrac{1}{2d}$
