6 3 ellipses

6-3 EllipsesObjectives:

1. Graph ellipses.

2. Find equations of ellipses.

Definition of an EllipseSet of all points P such that the sum

of the distances between P and the two foci (plural for “focus”) is constant.

d1 + d2 = constant

Characteristics of Ellipses

Standard Form Ellipse Equation: where a>b>0

The foci are on the major axis, c units from the center, where c2 = a2 – b2

Equation

Major Axis

Vertices

Co-Vertices

Horizontal (a, 0) (0, b)

Vertical (0, a) (b, 0)

Graphing Ellipses1. Write in standard form.2. Use a and b to plot vertices and co-

vertices.3. Sketch the ellipse.Example:Sketch the ellipse

and identify the foci.

Your Turn!Graph

and identify the foci.

Writing EquationsWrite an equation of the ellipse with

the given characteristics and center (0, 0).

Vertex: (0, 7) Co-Vertex: (-6, 0)

Vertex: (-4, 0)Focus: (2, 0)

Your Turn!Write an equation of the ellipse

with center (0,0) and:vertex at (3, 0)co-vertex at (0, -1)