Standard Equation of a Parabola

Definition. A parabola is the locus of a point in the plane that is equidistant from a fixed point (the focus) and a fixed line (the directrix) — this is exactly the focus–directrix definition of section 7.1.3 with $e=1$, since $SP=PM$ means the ratio $SP/PM$ is always $1$.

Deriving the standard equation $y^2=4ax$.

Let $S$ be the focus and $d$ the directrix. Draw $SZ$ perpendicular to the directrix (so $Z$ is the foot of that perpendicular, the point on $d$ closest to $S$), and let $O$ be the midpoint of segment $SZ$. By the very definition of the parabola, this midpoint $O$ is itself equidistant from $S$ and from the directrix — so $O$ lies on the parabola. This makes $O$ a natural choice of origin.

Set up coordinates with:

• $O$ as the origin,
• the line $OS$ as the $X$-axis,
• the line through $O$ perpendicular to $OS$ as the $Y$-axis.

Let $SZ = 2a$ (with $a>0$). Since $O$ is the midpoint of $SZ$, the focus is $S=(a,0)$ and the foot of the perpendicular is $Z=(-a,0)$. The directrix, being the vertical line through $Z$, has equation $x=-a$, i.e. $x+a=0$.

Now let $P(x,y)$ be any point on the parabola, and let $M$ be the foot of the perpendicular from $P$ to the directrix; since the directrix is $x=-a$, we have $M=(-a,y)$.

By the distance formula:

The defining property $SP=PM$ (since $e=1$) gives

Squaring both sides: …