Ellipse#

Definition: Ellipse

Let \(\lambda\) be a [[Planes|plane]] and let \(F_1\) and \(F_2\) be points from \(\lambda\).

The ellipse \(E\) with foci \(F_1\) and \(F_2\) is the [[Sets|subset]] of \(\lambda\) such that the sum of the distances between each point of \(E\) and the points \(F_1\) and \(F_2\) is constant.

\[ E = \{ P \in \lambda \mid d(P; F_1) + d(P; F_2) = \text{const} \} \]

The midpoint \(O\) of the [[Line Segments|line segment]] \(F_1F_2\) is known as \(E\)'s center.

The major axis of \(E\) is the [[Line Segments|line segment]] joining the two points of \(E\) which are the farthest away from \(O\).

The minor axis of \(E\) is the [[Line Segments|line segment]] connecting the two points of \(E\) which are the closest to \(O\).

Definition: Linear Eccentricity

The linear eccentricity of an [[Ellipse]] is the distance from its center to its foci.

Notation

The linear eccentricity is usually denoted by \(c\).

Theorem: Calculating Linear Eccentricity

The [[Ellipse|linear eccentricity]] of an [[Ellipse]] can be calculated using the length \(a\) of its semi-major axis and the length \(b\) of its semi-minor axis:

\[ c = \sqrt{a^2 - b^2} \]

Proof

TODO

Properties#

Theorem: Distance in Ellipses

For each point \(P\) of an [[Ellipse]] \(E\), the sum of the distances from \(P\) to the foci \(F_1\) and \(F_2\) is equal to the length of the major axis.

\[ d(P, F_1) + d(P, F_2) = 2a \]

Proof

TODO

Standard Equation#

Theorem: Standard Equation of an Ellipse

If \(E\) is whose center has coordinates \((0,0)\), then the coordinates \((x,y)\) of each point of \(E\) satisfy

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \]

where \(a\) is half the length of \(E\)'s [[Ellipse|major axis]] and \(b\) is half the length of \(E\)'s [[Ellipse|minor axis]].

Proof

TODO