10.4 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: 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)

Graphing1. Write in standard form.

2. Use a and b to plot vertices and co-vertices.

3. Draw the ellipse connecting all 4 points.

Example:

Draw the ellipse given by

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)

Area of an EllipseA = πab

Example:

A portion of the White House lawn is called The Ellipse. It is 1060 feet long and 890 feet wide.

1. Write an equation of The Ellipse.

2. Find the area of The Ellipse.

Modeling with an EllipseIn it’s elliptical orbit, Mercury ranges from

46.04 million km to 69.86 million km from the center of the sun, which is a focus of the orbit. Write an equation of the orbit.