Q.Find the equation of the tangent to the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ at the point $(a\sec\theta,\,b\tan\theta)$.

Find the tangent slope by implicit differentiation, evaluate at $(a\sec\theta,b\tan\theta)$, form the point-slope line, then clear denominators and use $\sec^2\theta-\tan^2\theta=1$ to reach the clean standard form.

Differentiating $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$:

$\dfrac{2x}{a^2}-\dfrac{2y\,y'}{b^2}=0\Rightarrow y'=\dfrac{b^2x}{a^2y}$

At $(a\sec\theta,b\tan\theta)$: $y'=\dfrac{b^2(a\sec\theta)}{a^2(b\tan\theta)}=\dfrac{b\sec\theta}{a\tan\theta}$

Tangent: $y-b\tan\theta=\dfrac{b\sec\theta}{a\tan\theta}(x-a\sec\theta)$

Multiply by $a\tan\theta$: …