Focal Distance and Latus Rectum of a Parabola

Two of the most useful numerical facts about a parabola — its focal distance formula and its latus-rectum length — follow quickly from the standard equation.

1. Focal distance formula. Let $P(x_1,y_1)$ be any point on $y^2=4ax$, and let $M=(-a,y_1)$ be the foot of the perpendicular from $P$ to the directrix $x+a=0$. By the very definition of the parabola, $SP=PM$. Computing $PM$ directly (it's a horizontal distance, since $M$ and $P$ share the $y$-coordinate):

So the focal distance of any point $P(x_1,y_1)$ on the parabola is simply

This is a genuinely useful shortcut: you never need the distance formula for a focal distance on a parabola — just add $a$ to the point's $x$-coordinate.

2. Length of the latus rectum. The latus rectum is the focal chord $LSL'$ perpendicular to the axis (section 7.1.4's definition, specialised to the parabola). By the curve's symmetry about the $X$-axis, the two halves $LS$ and $L'S$ are equal; call this common length $l$. Since $L$ lies directly above the focus $S=(a,0)$, its coordinates are $(a,l)$.

Because $L$ lies on the parabola, it must satisfy $y^2=4ax$:

Since $L$ is taken in the upper half (first quadrant), $l=2a$. So the length of the full latus rectum is

$LSL' = 2l = 2(2a) = 4a,$ …