Semi-Latus Rectum as Harmonic Mean

For any focal chord, the semi-latus rectum b²/a is the harmonic mean of the two focal distances r₁ and r₂ of its endpoints. Equivalently, 1/r₁ + 1/r₂ = 2a/b².

Let a focal chord have endpoints $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ on the ellipse, with $r_1 = a - ex_1$ and $r_2 = a - ex_2$ being their distances to the same focus (say $S'(c, 0)$).

Actually using the focus $S'$: $r_1 = a - ex_1$, $r_2 = a - ex_2$.

Since $P_1$ and $P_2$ are on opposite sides of the focus for a proper focal chord, $x_1$ and $x_2$ have the same or different signs depending on the chord. More cleanly:

$$\frac{1}{r_1} + \frac{1}{r_2} = \frac{1}{a-ex_1} + \frac{1}{a-ex_2}$$

For the focal chord through $S'(c,0)$, using the relation that both points satisfy $r = l/(1-e\cos\phi)$ (polar equation of ellipse) with angles $\phi$ and $\phi+\pi$:

$$r_1 = \frac{b^2/a}{1-e\cos\phi}, \quad r_2 = \frac{b^2/a}{1+e\cos\phi}$$ $$\frac{1}{r_1} + \frac{1}{r_2} = \frac{a}{b^2}(1-e\cos\phi) + \frac{a}{b^2}(1+e\cos\phi) = \frac{2a}{b^2}$$

The harmonic mean of $r_1$ and $r_2$:

$$\text{HM} = \frac{2r_1r_2}{r_1+r_2} = \frac{2}{\frac{1}{r_1}+\frac{1}{r_2}} = \frac{2}{\frac{2a}{b^2}} = \frac{b^2}{a}$$

The semi-latus rectum $b^2/a$ is the harmonic mean of the two focal segments of every focal chord — a fixed quantity independent of which focal chord is chosen.