Tangent to a Hyperbola

Definition (as before). A tangent to a hyperbola is a straight line intersecting the curve in two coincident points.

Finding the tangent at a point, using calculus. For $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$, differentiate implicitly:

At $P(x_1,y_1)$, slope $=\dfrac{b^2x_1}{a^2y_1}$. Point-slope form:

Dividing by $a^2b^2$: $\dfrac{xx_1}{a^2}-\dfrac{yy_1}{b^2}=\dfrac{x_1^2}{a^2}-\dfrac{y_1^2}{b^2}$. Since $P$ lies on the hyperbola, the right side is $1$. So the point-form tangent is

— again the "$x^2\to xx_1,\,y^2\to yy_1$" substitution pattern, now with the hyperbola's minus sign preserved. …