The ellipse: standard form, axes and foci

What an ellipse is

• An ellipse is the set of points for which the sum of the distances to two fixed points is constant.
• The two fixed points are the foci (singular: focus).
• It looks like a stretched circle — a circle is the special case where the two foci coincide.

Standard form centred at the origin

• An ellipse centred at the origin has equation:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

• $a$ — the semi-major axis (half the longer diameter) when $a > b$.
• $b$ — the semi-minor axis (half the shorter diameter).
• The major axis lies along the axis of the larger denominator.
• If $a > b$ the ellipse is wider than it is tall, with the major axis on the $x$-axis.

Vertices and axes

• Vertices on the major axis: $(\pm a, 0)$ (for $a > b$).
• Co-vertices on the minor axis: $(0, \pm b)$.
• The major axis has length $2a$; the minor axis has length $2b$.

Finding the foci

• The foci lie on the major axis, at $(\pm c, 0)$ (for $a > b$).
• Find $c$ from:

$c^2 = a^2 - b^2$

• $c$ — the distance from the centre to each focus.
• Note the subtraction: for an ellipse $c$ is always smaller than $a$.

Eccentricity

• The eccentricity $e$ measures how "stretched" the ellipse is:

$e = \frac{c}{a}, \qquad 0 \le e < 1$

• $e$ near $0$ — almost a circle.
• $e$ near $1$ — long and thin.