Equations of the Asymptotes

The two asymptotes of x²/a² − y²/b² = 1. The hyperbola approaches but never reaches them. Combined equation: x²/a² − y²/b² = 0. The asymptotes pass through the centre and make equal angles with both axes.

From $x^2/a^2 - y^2/b^2 = 1$, express $y$ for large $x$:

$$y^2 = b^2\!\left(\frac{x^2}{a^2} - 1\right) = \frac{b^2x^2}{a^2}\!\left(1 - \frac{a^2}{x^2}\right)$$ $$y = \pm\frac{bx}{a}\sqrt{1 - \frac{a^2}{x^2}}$$

As $x \to \infty$: $y \to \pm(bx/a)$. The hyperbola approaches $y = \pm(b/a)x$.

Distance from a point on the hyperbola to an asymptote:

Distance from $(a\sec\theta, b\tan\theta)$ to $bx - ay = 0$:

$$d = \frac{|b \cdot a\sec\theta - a \cdot b\tan\theta|}{\sqrt{a^2+b^2}} = \frac{ab|\sec\theta - \tan\theta|}{\sqrt{a^2+b^2}} \to 0 \text{ as } \theta \to 90°$$

This confirms the hyperbola approaches the asymptotes but never intersects them.

Combined equation of asymptotes: $y^2/b^2 - x^2/a^2 = 0$, equivalently $x^2/a^2 - y^2/b^2 = 0$.

Key relationship: $\text{Hyperbola} - \text{Asymptotes pair} =$ constant. Specifically:

$$\frac{x^2}{a^2} - \frac{y^2}{b^2} - 1 = 0 \quad \text{(hyperbola)}$$ $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 \quad \text{(asymptotes)}$$

The hyperbola and its asymptotes differ only by the constant on the right side. This is why they share the same centre and have the same asymptotic directions.

Angle between asymptotes:

$$\tan\alpha = \frac{b/a - (-b/a)}{1 + (b/a)(-b/a)} = \frac{2b/a}{1-b^2/a^2} = \frac{2ab}{a^2-b^2}$$

For a rectangular hyperbola ($a = b$): the angle between asymptotes is 90°.