The Ellipse

1. Core Concept Overview & Intuition

An ellipse is a conic section whose eccentricity is less than one. Equivalently, it is the locus of a point that moves such that the sum of its distances from two fixed points is constant. These two fixed points are called the foci of the ellipse. The constant sum of distances is equal to the length of the major axis.

The ellipse may also be described through a focus and a directrix: it is the locus of a point whose distance from a fixed point (the focus) is a constant fraction, less than one, of its distance from a fixed line (the directrix). That constant fraction is the eccentricity, denoted $e$, and for an ellipse $e<1$.

Geometrically, an ellipse is a closed oval curve with two perpendicular axes of symmetry. The longer axis is the major axis, of length $2a$, and the shorter is the minor axis, of length $2b$. The two axes meet at the center of the ellipse. The foci lie on the major axis, symmetrically placed about the center. When the center is at the origin and the major axis lies along the $x$-axis, the ellipse is widest in the horizontal direction and narrowest in the vertical direction. When the major axis lies along the $y$-axis, the orientation is reversed.

The shape of an ellipse is controlled by its eccentricity. A small eccentricity produces a curve close to a circle, while an eccentricity approaching one produces a long, thin ellipse. The latus rectum is the chord through a focus perpendicular to the major axis, and its length measures how "open" the curve is at the focus.

The study of the ellipse proceeds in two stages. First, the standard equation with center at the origin is established from the focus–directrix definition, together with the associated foci, vertices, directrices, eccentricity, and latus rectum. Second, the equation is translated so that the center lies at an arbitrary point $(h,k)$. From the equation of the curve, the tangent and normal lines at a given point are then derived, and the curve is also expressed in parametric form.

2. Mathematical Formalization & Core Definitions

2.1 Standard Equation of an Ellipse (Center at Origin)

Let the focus of the ellipse be $F$ and the directrix be the line $DM$. Let $V$ be the point where the ellipse meets the line through $F$ perpendicular to the directrix. By the focus–directrix property,

$$
FV=e\,VD
$$

which lies on the ellipse. Further away from $D$ along the line $DF$ there is another point $V'$ which satisfies

$$
FV'=e\,V'D.
$$

Let $C$ be the midpoint between $V$ and $V'$, and let the distance $CV=CV'=a$, so that $VV'=2a$. Then

$$
FV+FV'=e(VD+V'D)=e\,DD'
$$

where $V'D'=VD$ and $DD'=2CD$. Hence

$$
CF=ae.
$$

Place $C$ at the origin of the $XY$ plane, let $CD$ lie along the $x$-axis, and let the vertical line through $C$ be the $y$-axis. There is a second focus $F'$ and a second directrix $D'M'$ that would give the same curve. Writing $b^2=a^2(1-e^2)$, the equation of the ellipse reduces to the standard form

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

Here $a$ is the semi-major axis, $b$ is the semi-minor axis, and $e$ is the eccentricity. The relation

$$
b^2=a^2(1-e^2)
$$

links the three quantities.

2.2 Major and Minor Axes, Vertices, and Foci

The curve has two axes. The longer one is the major axis, which has length $2a$. The shorter axis, obtained when $x=0$, has length $2b$ and is the minor axis. The endpoints of the major axis are the vertices of the ellipse; the endpoints of the minor axis are the co-vertices. The foci lie on the major axis at a distance $ae$ from the center, so that for the origin-centered ellipse with major axis along the $x$-axis the foci are at $(\pm ae,0)$.

2.3 Directrices, Eccentricity, and Latus Rectum

The directrices are the two lines perpendicular to the major axis at distance $a/e$ from the center. For the origin-centered ellipse with major axis along the $x$-axis, the directrices are

$$
x=\pm\frac{a}{e}.
$$

The eccentricity is defined by

$$
e=\sqrt{1-\frac{b^2}{a^2}},
$$

and satisfies $0<e<1$. The latus rectum is the chord through a focus perpendicular to the major axis. For the origin-centered ellipse its length is

$$
\frac{2b^2}{a}.
$$

2.4 Translated Ellipses with Center $(h,k)$

When the center of the ellipse is $(h,k)$ rather than the origin, the equation is obtained by translating the standard form:

$$
\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1.
$$

The axes, vertices, foci, and directrices are all shifted by the same translation. The foci lie at $(h\pm ae,k)$ when the major axis is horizontal, and the directrices are the lines $x=h\pm a/e$. When the major axis is vertical, the roles of $a$ and $b$ are interchanged and the foci lie at $(h,k\pm ae)$.

The general equation of an ellipse may also be obtained directly from a focus and a directrix. Consider the ellipse whose focus is a given point and whose directrix is any line. Let $P(x,y)$ be any point on the ellipse. Then the focus–directrix condition gives

$$
(x-h)^2+(y-k)^2=e^2(\text{distance from }P\text{ to the directrix})^2,
$$

which expands to a general second-degree equation in $x$ and $y$. Completing the square in this general equation recovers the translated standard form and hence the center, axes, foci, eccentricity, and directrices.

2.5 Equations of Tangent and Normal to an Ellipse

Consider the ellipse

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

Let $P(x_1,y_1)$ be a point on the ellipse. The gradient of the chord from $P$ to a nearby point $Q$ on the ellipse is formed, and the limit is taken as $Q$ approaches $P$. Since both $P$ and $Q$ lie on the ellipse, their coordinates satisfy the equation of the ellipse; subtracting the two equations and taking the limit gives the gradient of the tangent at $P$. The resulting equation of the tangent at $P(x_1,y_1)$ is

$$
\frac{xx_1}{a^2}+\frac{yy_1}{b^2}=1.
$$

The normal at $P$ is perpendicular to the tangent at $P$. Its equation is

$$
\frac{a^2x}{x_1}-\frac{b^2y}{y_1}=a^2-b^2.
$$

2.6 Parametric Equations of an Ellipse

The standard ellipse can be described parametrically by

$$
x=a\cos\theta,\qquad y=b\sin\theta,
$$

where $\theta$ is the parameter (the eccentric angle). Substituting these expressions into the standard equation confirms that every such point lies on the ellipse, and conversely every point of the ellipse arises from some value of $\theta$. The parametric form is useful for locating points on the ellipse and for writing tangents and normals at a general point.