EllipsesLESSON 10.4

Objective

Define ellipses and parts of an ellipse

Graph ellipses

Find the equation of an ellipse

Ellipses

An ellipse is the set of all points on a plane whose distance from two fixed focal points (foci) add up to

a constant number.

Ellipses

A line drawn through an ellipse containing the foci is

called the major axis. The intersections between the major axis and the ellipse are the vertices 𝑉1 and 𝑉2.

Ellipses

𝑥−ℎ 2

𝑎2+

𝑦−𝑘 2

𝑏2= 1 major axis: ↔

𝑥−ℎ 2

𝑏2+

𝑦−𝑘 2

𝑎2= 1 major axis: ↕

*NOTE: 𝑎 > 𝑏

Ellipses

Identify the center of the ellipse, 𝑎, and 𝑏.

1) 𝑥2

9+

𝑦2

49= 1 2)

𝑥−1 2

25+

𝑦−4 2

9= 1

Ellipses

Identify the center of the ellipse, 𝑎, and 𝑏.

3) 𝑥−1 2

4+

𝑦+2 2

16= 1 4) 𝑥 − 2 2 +

𝑦2

4= 1

Ellipses

The major axis has two points that intersect the

ellipse. These points are vertices. Vertices can be found by adding and subtracting 𝑎 from the center.

Ellipses

Identify the center, 𝑎, and the vertices.

5) 𝑥2

16+

𝑦2

36= 1 6)

𝑥2

16+ 𝑦2 = 1

Ellipses

Identify the center, 𝑎, and the vertices.

7) 𝑥+1 2

36+

𝑦+1 2

25= 1

Ellipses

The focal points (foci) of an ellipse are a set distance 𝑐 away from the center along the major

axis. For either major axis (↔, ↕)

𝑏2 = 𝑎2 − 𝑐2 or 𝑐 = 𝑎2 − 𝑏2

Ellipses

Identify the foci.

8) 𝑥2

9+

𝑦2

25= 1 9)

𝑥2

16+ 𝑦2 = 1

Ellipses

Identify the center, vertices, and foci. GRAPH.

10) 𝑥2

9+

𝑦2

25= 1 11)

𝑥2

100+

𝑦2

64= 1

Ellipses

Identify the center, vertices, and foci. GRAPH.

12) 𝑥+2 2

25+

𝑦2

16= 1

Ellipses

Use the information provided to write the standard

form equation of the ellipse.

13) Center: (0,0)

Vertex: (6,0)

Focus: (3,0)

Ellipses

Use the information provided to write the standard

form equation of the ellipse.

13) Foci: (0,6), (0, −6)

Length of major axis: 20