Chord with a Given Midpoint

Chord of x²/a² − y²/b² = 1 whose midpoint is (x₁, y₁). Explicitly: xx₁/a² − yy₁/b² − 1 = x₁²/a² − y₁²/b² − 1.

Let $(x_1, y_1)$ be the midpoint of a chord of $x^2/a^2 - y^2/b^2 = 1$ with endpoints $(x_2, y_2)$ and $(x_3, y_3)$.

Both on the hyperbola:

$$\frac{x_2^2-x_3^2}{a^2} - \frac{y_2^2-y_3^2}{b^2} = 0$$ $$\frac{(x_2+x_3)(x_2-x_3)}{a^2} = \frac{(y_2+y_3)(y_2-y_3)}{b^2}$$

Using midpoint: $x_2+x_3 = 2x_1$, $y_2+y_3 = 2y_1$:

$$\text{Slope of chord} = \frac{y_2-y_3}{x_2-x_3} = \frac{b^2x_1}{a^2y_1}$$

(Note: positive slope for the hyperbola, compared to negative for the ellipse — the minus sign in the hyperbola equation reverses the sign.)

Chord through $(x_1,y_1)$ with this slope:

$$y - y_1 = \frac{b^2x_1}{a^2y_1}(x-x_1)$$

Rearranging: $T = S_1$, where $T = xx_1/a^2 - yy_1/b^2 - 1$ and $S_1 = x_1^2/a^2 - y_1^2/b^2 - 1$.