Length of Latus Rectum

The latus rectum is the chord through the focus perpendicular to the axis. Its endpoints are (a, 2a) and (a, −2a). Half the latus rectum (semi-latus rectum) = 2a.

For $y^2 = 4ax$, the latus rectum passes through the focus $S(a, 0)$ and is perpendicular to the axis, so it lies on the line $x = a$.

Substitute $x = a$ into $y^2 = 4ax$:

$$y^2 = 4a \cdot a = 4a^2 \implies y = \pm 2a$$

The endpoints of the latus rectum are $(a, 2a)$ and $(a, -2a)$.

Length of latus rectum $= 2a - (-2a) = 4a$.

Semi-latus rectum: $2a$ — this is the focal distance of either endpoint:

$$PS = x + a = a + a = 2a \checkmark$$

Significance in focal chord theory: The latus rectum is the shortest focal chord. Any other focal chord is longer. This follows because the length of a focal chord with parameter $t$ is $a(t + 1/t)^2 \geq 4a$ (by AM-GM), with equality when $|t| = 1$ (the latus rectum endpoints).