Hyperbola

A hyperbola is the locus with $e>1$: its distance from the focus is greater than its distance from the directrix, scaled by $e$.

(i) Standard form. With the same construction as the ellipse (points $A,A'$ dividing $SZ$ internally/externally in ratio $e:1$, $AA'=2a$, $C$ the midpoint as origin) but now $e>1$, working through $AS=eAZ$, $A'S=eA'Z$ gives again $CS=ae$ and $CZ=a/e$, so focus $S(ae,0)$, directrix $x=a/e$. Applying $SP^2=e^2PM^2$ and simplifying, with $b^2=a^2(e^2-1)$ (positive since $e>1$), gives

the hyperbola in standard form, symmetric about both axes, with a second focus $S'(-ae,0)$ and matching directrix $x=-a/e$ by symmetry, exactly as for the ellipse.

Vocabulary (Definition 5.5). $AA'$ (length $2a$, $A(a,0),A'(-a,0)$ the two vertices) is the transverse axis; $BB'$ (length $2b$, where $B(0,b)$ is not on the curve) is the conjugate axis. Taking $ae=c$, so $b^2=c^2-a^2$: foci $(\pm c,0)$, directrices $x=\pm a/e$, and (by the same substitution as for the ellipse) latus rectum $=2b^2/a$ — proved directly in Ex. 5.2 Q6.

Key difference-of-focal-distances property (Ex. 5.2 Q7). For any point $P$ on the hyperbola, $|PS-PS'|=2a$ (constant) — the hyperbola's analogue of Theorem 5.5, proved the same way from $PS=ex-a$, $PS'=ex+a$ on the right branch (and symmetrically on the left).

Asymptotes. As a point on the curve moves further from the centre, the hyperbola's two branches approach, but never touch, two straight lines called asymptotes — a feature the parabola and ellipse do not have. The circle described on the transverse axis as diameter, $x^2+y^2=a^2$, is again called the auxiliary circle (used for parametrising the hyperbola, §5.5.1). …