Q.The coordinate of the vertices of the rectangular hyperbola $xy = 16$ are :

Setting $e>1$ in the conic definition gives the hyperbola — its distance from the focus is greater than $e$ times its distance from the directrix.

Standard form: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$, where $b^2=a^2(e^2-1)$. Foci $(\pm c,0)$ with $c^2=a^2+b^2$ (equivalently $c=ae$); vertices $(\pm a,0)$; directrices $x=\pm a/e=\pm a^2/c$. Transverse axis $2a$ (joins the vertices), conjugate axis $2b$ (does not meet the curve); latus rectum $=2b^2/a$ (same formula shape as the ellipse — proved by substituting $x=ae=c$).

Key difference property. For any point $P$ on the hyperbola, $|PS-PS'|=2a$ (constant) — the hyperbola's analogue of the ellipse's focal-distance-sum property.

Asymptotes. Two straight lines the branches approach but never touch as $|x|\to\infty$ — a feature unique to the hyperbola among the non-degenerate conics. …