Ellipses as Geometric Loci

Definition Let $F_1$ and $F_2$ be points of the plane, let $2c = \lvert F_1F_2 \rvert$ be the distance between them, and let $a$ be a number with $a > c \ge 0$. The ellipse with foci $F_1$ and $F_2$ and constant sum $2a$ is the set of all points $P$ of the plane with

Points where the sum is less than $2a$ lie inside the ellipse, and points where it is greater lie outside. The condition $a > c$ is needed. By the triangle inequality the sum is never less than $2c$, so a sum of exactly $2c > 0$ gives only the segment $F_1F_2$, and a sum below $2c$ gives no points at all.

Definition The foci (singular focus) of an ellipse are the two fixed points $F_1$ and $F_2$ of its definition. When they are different points, the line through them is called the focal axis. Both foci lie inside the ellipse, since at a focus the sum of distances is $0 + 2c < 2a$. The ellipse meets the focal axis in exactly two points, one beyond each focus.

Definition The center of an ellipse is the midpoint of the segment $F_1F_2$, so each focus lies at distance $c$ from the center. The ellipse is symmetric about its center. If $P$ is on the ellipse and $P'$ is the point with the center as the midpoint of $PP'$, then $P'$ is on the ellipse too, because the half turn about the center swaps the foci and keeps every distance. The ellipse is also symmetric about the focal axis and about the line through the center perpendicular to it.

Definition Suppose the foci are different points, so that the focal axis exists. The vertices of the ellipse are the two points where it meets the focal axis, and the segment between them is the major axis. For the vertex $V$ beyond $F_2$ we have $\lvert VF_1 \rvert = \lvert VF_2 \rvert + 2c$, so the sum condition gives $\lvert VF_2 \rvert = a - c$, and $V$ lies $c + (a - c) = a$ units from the center. The same holds for the other vertex, so the major axis has length $2a$, the constant sum. The number $a$ is the semimajor axis. The covertices are the two points where the ellipse meets the line through the center perpendicular to the focal axis, and the segment between them is the minor axis. A covertex is equally far from both foci, so each of those distances is $a$, and the Pythagorean theorem puts it at distance

from the center. The number $b$ is the semiminor axis, and $0 < b < a$. When the foci coincide, $c = 0$ and the ellipse is a circle of radius $a = b$, so every diameter can serve as either axis.

Definition For every ellipse with semimajor axis $a$, semiminor axis $b$ and center to focus distance $c$,

This is the Pythagorean theorem in the right triangle with vertices at the center, a focus and a covertex. Its legs are $c$ and $b$, and its hypotenuse, from the covertex to the focus, has length $a$. Any two of the three numbers determine the third.

Definition Put the center at the origin and the foci at $(\pm c, 0)$, with $a > c \ge 0$ and $b = \sqrt{a^2 - c^2}$. A point $(x, y)$ lies on the ellipse exactly when

To derive it, write the sum condition with the distance formula, isolate one square root and square. After simplifying this leaves

Squaring again and dividing by $a^2b^2$ gives the equation. Squaring can create extra solutions in general, but not here. Every point of the equation has $\lvert x \rvert \le a$, and its distances to $F_1 = (-c, 0)$ and $F_2 = (c, 0)$ are

Both are positive, since $\frac{c}{a}\lvert x \rvert \le c < a$, and they add up to $2a$. With the foci at $(0, \pm c)$ the letters trade places and the equation is

When $c > 0$, the larger denominator in both forms is $a^2$, and the foci lie on the axis of the variable over it. Equal denominators mean $c = 0$ and a circle.

Definition The eccentricity of an ellipse is the ratio

It depends only on the shape and not on the size, since enlarging the ellipse multiplies $a$ and $c$ by the same factor. It also fixes the ratio of the axes, because

Definition When the two foci coincide at a point $C$, we have $c = 0$, so $b = a$ and $e = 0$. The sum condition becomes $2\lvert PC \rvert = 2a$, that is $\lvert PC \rvert = a$, which is the definition of the circle with center $C$ and radius $a$. With $C$ at the origin the standard equation becomes $x^2 + y^2 = a^2$. So every circle is an ellipse, and an ellipse with $c > 0$ is never a circle, because its vertices are farther from the center than its covertices.