Tangent at a Point on the Hyperbola

Tangent to x²/a² − y²/b² = 1 at point (x₁, y₁) lying on it. Obtained by the T = 0 rule: replace x² → xx₁, y² → yy₁.

Let $P(x_1, y_1)$ lie on $x^2/a^2 - y^2/b^2 = 1$, so $x_1^2/a^2 - y_1^2/b^2 = 1$.

Differentiating implicitly:

$$\frac{2x}{a^2} - \frac{2y}{b^2}\frac{dy}{dx} = 0 \implies \frac{dy}{dx} = \frac{b^2x}{a^2y}$$

Slope at $P$: $m = b^2x_1/(a^2y_1)$.

Tangent through $P$:

$$y - y_1 = \frac{b^2x_1}{a^2y_1}(x-x_1)$$ $$a^2y_1(y-y_1) = b^2x_1(x-x_1)$$ $$b^2x_1x - a^2y_1y = b^2x_1^2 - a^2y_1^2 = a^2b^2\!\left(\frac{x_1^2}{a^2} - \frac{y_1^2}{b^2}\right) = a^2b^2$$

Dividing by $a^2b^2$:

$$\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1$$

Note the sign: The tangent to the hyperbola has a minus sign between the two terms, matching the hyperbola equation itself. The T = 0 rule (replace $x^2 \to xx_1$, $y^2 \to yy_1$) still applies universally.