Other Standard Forms of a Parabola

The derivation in section 7.1.5 assumed the parabola opens to the right, with focus on the positive $X$-axis. But exactly the same reasoning, with the coordinate axes chosen differently, produces three more "standard" forms — one for each of the four possible opening directions:

In every case $a>0$ (or $b>0$) is the distance from the vertex to the focus, and every one of the results from sections 7.1.6–7.1.7 (symmetry, focus, directrix, latus rectum, focal distance) carries across with the appropriate sign and axis swapped. The comparison table below (also reproduced as a note on this section) is the single most useful reference for solving "identify focus/directrix/latus rectum" problems, since it lets you read off every property the moment you have matched a given equation to one of these four forms:

• $y^2=4ax$: focus $(a,0)$; directrix $x+a=0$; latus-rectum end points $(a,\pm2a)$; length $|4a|$; axis is the $X$-axis ($y=0$); tangent at vertex is the $Y$-axis; focal distance of $(x_1,y_1)$ is $|x_1+a|$.
• $x^2=4by$: focus $(0,b)$; directrix $y+b=0$; latus-rectum end points $(\pm2b,b)$; length $|4b|$; axis is the $Y$-axis ($x=0$); tangent at vertex is the $X$-axis; focal distance of $(x_1,y_1)$ is $|y_1+b|$. …