Length of Latus Rectum

Each focus has a latus rectum passing through it perpendicular to the major axis. Endpoints of the latus rectum through (c, 0) are (c, ±b²/a). The semi-latus rectum b²/a is the harmonic mean of the two focal segments of any focal chord.

The latus rectum passes through the focus $S'(c, 0)$ and is perpendicular to the major axis, so it lies on $x = c$.

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

$$\frac{c^2}{a^2} + \frac{y^2}{b^2} = 1 \implies y^2 = b^2\!\left(1 - \frac{c^2}{a^2}\right) = \frac{b^2(a^2-c^2)}{a^2} = \frac{b^4}{a^2}$$ $$y = \pm \frac{b^2}{a}$$

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

Semi-latus rectum $= b^2/a$ — this is also the focal distance of either endpoint of the latus rectum:

$$r = a - e \cdot c = a - \frac{c}{a} \cdot c = a - \frac{c^2}{a} = \frac{a^2 - c^2}{a} = \frac{b^2}{a} \checkmark$$

Harmonic mean property: For any focal chord with focal segments $r_1$ and $r_2$:

$$\frac{2}{\frac{1}{r_1} + \frac{1}{r_2}} = \frac{2r_1r_2}{r_1+r_2} = \frac{b^2}{a} \quad \text{(semi-latus rectum)}$$

The latus rectum is the unique focal chord for which both focal segments are equal to $b^2/a$.