Tangent at Parametric Point θ

Tangent at (a secθ, b tanθ). Slope = (b secθ)/(a tanθ) = b/(a sinθ).

Substitute $x_1 = a\sec\theta$, $y_1 = b\tan\theta$ into $xx_1/a^2 - yy_1/b^2 = 1$:

$$\frac{x \cdot a\sec\theta}{a^2} - \frac{y \cdot b\tan\theta}{b^2} = 1$$ $$\frac{x\sec\theta}{a} - \frac{y\tan\theta}{b} = 1$$

Slope: From $y = (bx\sec\theta - b)/(a\tan\theta)$, slope $= b\sec\theta/(a\tan\theta) = b/(a\sin\theta)$.

Tangent parallel to asymptote $y = (b/a)x$: Asymptote slope $= b/a$. Setting tangent slope $= b/a$:

$$\frac{b}{a\sin\theta} = \frac{b}{a} \implies \sin\theta = 1 \implies \theta = 90°$$

But $\theta = 90°$ is excluded (the parametrisation is undefined there). This confirms that no tangent to the hyperbola is parallel to the asymptotes — the asymptotes themselves are the limiting tangents.

Two tangents from an external point: For each external point, two tangents exist. For a point between the asymptotes (on the same branch), one tangent exists. A point on the asymptote gives a tangent that is the asymptote itself.