The Four Standard Orientations and the Latus Rectum

The derivation above assumed the parabola opens to the right, with the focus to the right of the vertex. Three mirror-image choices are just as natural, and together the four cover every parabola whose axis is parallel to a coordinate axis and whose vertex is the origin:

(In every row $a>0$, and the vertex is the origin.) Each equation follows from exactly the same distance argument as $y^2=4ax$, just with the focus placed on the other side, or on the $y$-axis instead of the $x$-axis. If the vertex is not the origin but some point $(h,k)$, and the axis is still parallel to a coordinate axis, translating the origin to $(h,k)$ turns each of the four equations above into its shifted version — for instance $y^2=4ax$ becomes $(y-k)^2=4a(x-h)$ for a rightward-opening parabola with vertex $(h,k)$.

Two more pieces of vocabulary are used constantly. A chord of the parabola is any line segment joining two points on it; a chord through the focus is a focal chord; and a chord through a point $P$ that is perpendicular to the axis is called the double ordinate of $P$. The double ordinate that happens to pass through the focus is special enough to have its own name: the latus rectum. Its length is easy to pin down for $y^2=4ax$: the latus rectum meets the curve where $x=a$ (the focus's $x$-coordinate), so $y^2=4a\cdot a=4a^2$, giving $y=\pm 2a$. The two endpoints are therefore $(a,2a)$ and $(a,-2a)$, a distance of $4a$ apart — so the length of the latus rectum is $4a$ for every one of the four standard forms. Knowing the latus rectum's length is often the fastest way to pin down $a$, and hence the whole equation, from a word problem. …