Normal at a Point on the Hyperbola

Normal to x²/a² − y²/b² = 1 at (x₁, y₁). Note the + signs on both terms (contrast with the ellipse normal which has a − sign).

At $P(x_1, y_1)$ on $x^2/a^2 - y^2/b^2 = 1$, the tangent slope is $b^2x_1/(a^2y_1)$.

Normal slope: $-a^2y_1/(b^2x_1)$.

Normal through $P$:

$$y - y_1 = -\frac{a^2y_1}{b^2x_1}(x-x_1)$$ $$b^2x_1(y-y_1) = -a^2y_1(x-x_1)$$ $$a^2y_1x + b^2x_1y = x_1y_1(a^2+b^2)$$

Dividing by $x_1y_1$:

$$\frac{a^2x}{x_1} + \frac{b^2y}{y_1} = a^2+b^2$$

Comparison with ellipse normal: Ellipse normal: $a^2x/x_1 - b^2y/y_1 = a^2-b^2$. Hyperbola normal: $a^2x/x_1 + b^2y/y_1 = a^2+b^2$. Both the sign between terms and the right-hand side differ.

For the ellipse, $a^2-b^2 = c^2$. For the hyperbola, $a^2+b^2 = c^2$ as well. So both can be written as $(\cdots) = c^2$, but the left sides differ.