$\text{ If }\alpha +\beta =\pi \text{ then the chord joining the points }\alpha \text{ and }\beta \text{ for the hyperbola }\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\text{ passes through }$

$\text{ Points on hyperbola }P(a\sec \alpha ,b\tan \alpha )\text{ and }Q(a\sec \beta ,b\tan \beta )$

$\beta =\pi -\alpha$

$\therefore Q(a\sec (\pi -\alpha ),b\tan (\pi -\alpha \left)\right)\text{ or }Q(-a\sec \alpha ,-b\tan \alpha )$

$\text{Clearly points P and Q are symmetric about origin. Hence chord joining P and Q passes through origin.}$