multiply and divide rational...

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