Parabola

Since $e=1$ for a parabola, a parabola is simply the set of points equidistant from a fixed focus and a fixed directrix.

(i) Standard form, vertex at the origin. Let $S$ be the focus and $\ell$ the directrix; draw $SZ\perp\ell$, and take the line $SZ$ (produced) as the $x$-axis with the perpendicular bisector of $SZ$ as the $y$-axis, so the origin $O$ is their intersection. Let $SZ=2a$, so $S=(a,0)$ and $\ell:x+a=0$. For a moving point $P(x,y)$ with $PM\perp\ell$, the defining condition $SP=PM$ (since $e=1$) gives

the parabola in standard form. The three other origin-vertex orientations follow by symmetry: $y^2=-4ax$ (opens left), $x^2=4ay$ (opens up), $x^2=-4ay$ (opens down).

Vocabulary (Definition 5.3). The line through the focus perpendicular to the directrix is the axis; where the axis meets the curve is the vertex; any chord through the focus is a focal chord; the focal chord perpendicular to the axis is the latus rectum, and its endpoints for $y^2=4ax$ are $(a,\pm2a)$ (found by substituting $x=a$), so its length is $4a$. The parabola $y^2=4ax$ is symmetric about the $x$-axis (replacing $y$ by $-y$ leaves the equation unchanged) and lies entirely on the side $x\ge0$.

(ii) Vertex at $(h,k)$. Shifting the origin to $(h,k)$: when the axis is parallel to the $x$-axis, the equation is $(y-k)^2=\pm4a(x-h)$; when parallel to the $y$-axis, $(x-h)^2=\pm4a(y-k)$. Summarised (with $a>0$ throughout):