Recently, I learnt that other than using a quadratic equation to describe a parabola, it is also possible to say that a parabola is the locus of points that is equidistant from a focus and a directrix.
The idea of the parabola having points equidistant from a single focus and a line (directrix) seems like an acceptable idea, but for some reason,
I've been wondering how we know that this definition works? Is there a way to show that all the points are equidistant from the focus and directrix?
First, make a change or coordinates (translation and rotation) to get an equation $y=ax^2$ for your parabola. Take an unknown focus $F$ as $(0,b)$ and a directrix as $y=-b$: it must be $-b$ because for $x=0$, the point $(0,0)$ on the parabola is equidistant from $(0,b)$ and the directrix. Let $M(x,ax^2)$ be a moving point on the parabola, and $H(x,-b)$ the orthogonal projection on the directrix.
Now, let's check the distances, or equivalently the squares of the distances:
$$MH^2=(ax^2+b)^2=a^2x^4+2abx^2+b^2$$ $$MF^2=x^2+(ax^2-b)^2=a^2x^4+(1-2ab)x^2+b^2$$
They are equal if $2ab=1-2ab$, that is $b=\dfrac{1}{4a}$.