Ellipse

An ellipse is the locus with $0<e<1$: its distance from the focus is less than its distance from the directrix, scaled by $e$.

(i) Standard form. Let $S$ be a focus, $\ell$ a directrix, $e\in(0,1)$. Let $A,A'$ divide $SZ$ (where $SZ\perp\ell$) internally and externally in the ratio $e:1$, with $AA'=2a$ and $C$ (their midpoint) the origin; $CZ$ is the $x$-axis. Working through $\dfrac{SA}{AZ}=\dfrac{SA'}{A'Z}=e$ gives $CZ=\dfrac ae$ and $CS=ae$, so the focus is $S(ae,0)$ and the directrix is $x=\dfrac ae$. Applying $SP^2=e^2PM^2$ to a moving point $P(x,y)$ and simplifying (using $b^2=a^2(1-e^2)$, a positive quantity since $e<1$) gives

the ellipse in standard form, symmetric about both axes. By symmetry there is a second focus $S'(-ae,0)$ with matching directrix $x=-\dfrac ae$ — unlike the parabola, the ellipse (and, similarly, the hyperbola) has two foci and two directrices.

Vocabulary (Definition 5.4). $AA'$ (length $2a$) is the major axis; $BB'$ (length $2b$, where $B(0,b)$) is the minor axis; $a,b$ are the semi-major/semi-minor axes. Taking $ae=c$, so $b^2=a^2-c^2$: the vertices are $(\pm a,0)$, foci $(\pm c,0)$, directrices $x=\pm a/e$. Substituting $x=ae$ into the equation, as in the focus-side latus rectum, gives $y=\pm b^2/a$, so the latus rectum has length $2b^2/a$.

Theorem 5.5 (sum of focal distances). For any point $P$ on the ellipse, $SP+S'P=2a$ (constant — in fact this is often taken as the defining "string-and-two-pins" construction of an ellipse). Proof sketch: using the focus-directrix relation on each focus separately, $SP=a-ex$ and $S'P=a+ex$, so $SP+S'P=2a$ regardless of $x$.

Remarks. As $e\to0$, $b\to a$ and the ellipse rounds into a circle (the directrix recedes to infinity, and $SP/PM\to0$, i.e. $PM\to\infty$ — which is exactly what "directrix at infinity" means); this is why the circle is sometimes called the $e=0$ degenerate ellipse. The circle on the major axis as diameter, $x^2+y^2=a^2$, is the auxiliary circle (used to parametrise the ellipse, §5.5.1); the circle on the minor axis as diameter, $x^2+y^2=b^2$, is the incircle. …