Standard Equation and Properties of an Ellipse

An ellipse is defined as the locus of a point $P$ that moves so that the sum of its distances from two fixed points, the foci $F_1$ and $F_2$, is always the same constant, $2a$ (with $2a$ greater than the distance $F_1F_2=2c$ between the foci, so the locus is a genuine closed curve and not empty).

Standard equation. Placing the foci symmetrically on the $x$-axis at $F_1(-c,0)$ and $F_2(c,0)$, with the midpoint of $F_1F_2$ at the origin, the condition $PF_1+PF_2=2a$ works out (after two applications of the distance formula and simplification) to the standard equation of an ellipse:

Here $a$ is the semi-major axis and $b$ the semi-minor axis; the major axis (length $2a$) always lies along the axis containing the foci, and is always the longer of the two axes, so $a>b$ always.

Eccentricity. The eccentricity of the ellipse is defined as $e=\dfrac{c}{a}$, and since $0<c<a$ for a genuine ellipse, $0<e<1$ always. Rearranging $b^2=a^2-c^2$ using $c=ae$ gives the equivalent, frequently-used relation

The closer $e$ is to $0$, the closer the ellipse is to a circle (indeed $e=0$ exactly recovers the circle, $a=b$); the closer $e$ is to $1$, the more elongated the ellipse becomes.

Key terms. The foci are $F_1(-c,0)$ and $F_2(c,0)$, where $c=ae$; the vertices are the endpoints of the major axis, $(\pm a,0)$; the major axis has length $2a$ and the minor axis (along the $y$-axis, between $(0,-b)$ and $(0,b)$) has length $2b$; the centre is the origin, the midpoint of both axes. The latus rectum through either focus, perpendicular to the major axis, has length

$\text{latus rectum}=\frac{2b^2}{a}.$ …