Parametric Form of xy = c²

Every point on xy = c² is (ct, c/t) for a unique t ∈ ℝ\{0}. t > 0 gives the first quadrant branch; t < 0 gives the third quadrant branch.

Set $x = ct$, $y = c/t$ for $t \neq 0$. Then:

$$xy = ct \cdot \frac{c}{t} = c^2 \checkmark$$

Every point on $xy = c^2$ corresponds to a unique $t \in \mathbb{R}\setminus\{0\}$.

Branch identification: $t > 0$ gives first quadrant ($x, y > 0$); $t < 0$ gives third quadrant ($x, y < 0$).

Slope at $(ct, c/t)$: From $y = c^2/x$, $dy/dx = -c^2/x^2 = -(c/t)^2/(ct)^2 \cdot (ct)^2/c^2$:

$$\frac{dy}{dx} = -\frac{c^2}{x^2} = -\frac{c^2}{c^2t^2} = -\frac{1}{t^2}$$

Chord joining $t_1$ and $t_2$: The chord from $(ct_1, c/t_1)$ to $(ct_2, c/t_2)$ has slope:

$$m = \frac{c/t_1 - c/t_2}{ct_1 - ct_2} = \frac{c(t_2-t_1)/(t_1t_2)}{c(t_1-t_2)} = -\frac{1}{t_1t_2}$$

Chord equation: $x + t_1t_2 y = c(t_1+t_2)$.

Midpoint of chord: $\left(\frac{c(t_1+t_2)}{2},\; \frac{c(1/t_1+1/t_2)}{2}\right) = \left(\frac{c(t_1+t_2)}{2},\; \frac{c(t_1+t_2)}{2t_1t_2}\right)$.