Length of Latus Rectum

Each focus has a latus rectum of length 2b²/a, the same formula as the ellipse and parabola. Endpoints of the latus rectum through (c, 0) are (c, ±b²/a).

The latus rectum passes through the focus $S'(c, 0)$ perpendicular to the transverse axis, on the line $x = c$.

Substitute $x = c$ into $x^2/a^2 - y^2/b^2 = 1$:

$$\frac{c^2}{a^2} - \frac{y^2}{b^2} = 1 \implies \frac{y^2}{b^2} = \frac{c^2}{a^2} - 1 = \frac{c^2-a^2}{a^2} = \frac{b^2}{a^2} \cdot \frac{b^2}{b^2}$$

Wait: $c^2 - a^2 = b^2$, so:

$$\frac{y^2}{b^2} = \frac{b^2}{a^2} \implies y^2 = \frac{b^4}{a^2} \implies y = \pm\frac{b^2}{a}$$

Length of latus rectum $= \dfrac{2b^2}{a}$.

This is identical in form to the ellipse latus rectum. The latus rectum of any conic section has length $2b^2/a$ (for parabola: $b =$ undefined, but $2b^2/a \to 4a$ when interpreted correctly).

Focal distance of each endpoint: $r = ex_1 - a = e \cdot c - a = ec - a$. Using $c = ae$: $r = ae^2 - a = a(e^2-1) = b^2/a$. So the semi-latus rectum is $b^2/a$, equal to the focal distance of the latus rectum endpoint.