Length of a Focal Chord

Length of the focal chord with parameter t at one end (and −1/t at the other). Minimum value 4a (the latus rectum) is achieved when t = ±1.

For a focal chord with endpoints $P_1(at^2, 2at)$ and $P_2(a/t^2, -2a/t)$ (using $t_1 = t$, $t_2 = -1/t$):

Focal distances:

$$P_1S = at^2 + a = a(t^2 + 1)$$ $$P_2S = \frac{a}{t^2} + a = a\!\left(\frac{1}{t^2} + 1\right) = \frac{a(t^2+1)}{t^2}$$

Length of focal chord:

$$\ell = P_1S + P_2S = a(t^2+1) + \frac{a(t^2+1)}{t^2} = a(t^2+1)\!\left(1 + \frac{1}{t^2}\right) = a\cdot\frac{(t^2+1)^2}{t^2}$$ $$\ell = a\!\left(t + \frac{1}{t}\right)^{\!2}$$

Minimum length: By AM-GM, $t + 1/t \geq 2$ for $t > 0$ (and $\leq -2$ for $t < 0$), so $(t+1/t)^2 \geq 4$:

$$\ell_{\min} = 4a \quad \text{(the latus rectum, at } |t| = 1\text{)}$$

Harmonic mean property: The semi-latus rectum $2a$ is the harmonic mean of $P_1S$ and $P_2S$:

$$\frac{2}{\frac{1}{P_1S} + \frac{1}{P_2S}} = \frac{2 \cdot P_1S \cdot P_2S}{P_1S + P_2S} = \frac{2a(t^2+1) \cdot a(t^2+1)/t^2}{a(t^2+1)^2/t^2} = 2a \checkmark$$