Study Notes on Ellipses

Introduction to Ellipse

An ellipse resembles an oval and can be formed when a plane intersects a right circular cone.

An ellipse is defined as a set of coplanar points where the sum of distances from two fixed points (foci) is constant.

Key Concepts

• Foci: Fixed points of the ellipse.

• Major Axis: The longest axis of the ellipse that passes through the foci.

• Minor Axis: The shortest axis that is perpendicular to the major axis.

• Vertices: Endpoints of the major axis.

• Co-vertices: Endpoints of the minor axis.

• Center: The midpoint between the two foci and vertices.

• Directrix: A line parallel to the minor axis at a distance from the vertices equal to the distance from the focus.

Standard Form of Ellipse Equation

1. Ellipse with foci on the x-axis:
   $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $c = \sqrt{a^2 - b^2}$ and $a > b$.

2. Ellipse with foci on the y-axis:
   $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ where $c = \sqrt{a^2 - b^2}$ and $a > b$.

Deriving the Equation of an Ellipse

• For a centered ellipse at the origin:

  • foci: $F1(c, 0)$ and $F2(-c, 0)$,

  • sum of distances from a point $P(x, y)$ on the ellipse to the foci is $2a$.

Examples

• Example 1: Vertices (±8, 0), foci (±5, 0)

  • Major axis = x-axis.

  • $\frac{x^2}{64} + \frac{y^2}{39} = 1$

• Example 2: Vertices (0, ±9), foci (0, ±4)

  • Major axis = y-axis.

  • $\frac{x^2}{65} + \frac{y^2}{81} = 1$

• Continue with similar examples to demonstrate how to substitute vertices and foci into the equations to derive the standard form.