10 – 4 Ellipses

Ellipse

Center (0, 0)

Center (0, 0)

Center (0, 0)

Writing an Equation

• What is an equation in standard form of an ellipse centered at the origin with vertex (-6, 0) and co-vertex (0, 3)?

• What is an equation in standard form of an ellipse centered at the origin with vertex (0, 5) and co-vertex (2, 0)?

• What is an equation in standard form of an ellipse centered at the origin with vertex (8, 0) and co-vertex (0, 5)?

Finding the Foci

• Use c2 = a2 – b2 to find the foci of ±c

• Vertical ellipse: (0, ±c)

• Horizontal ellipse (±c, 0)

• What are the foci of the ellipse with the equation 25x2 + 9y2 = 225

• What are the foci of the ellipse with the equation 36x2 + 100y2 = 3600

• Like parabolas, ellipses have an important reflexive property related to the foci

• Any line emanating from one focus of an ellipse will reflect off the ellipse directly onto the other focus

Using the Foci

• A room with an elliptical ceiling forms a “whispering gallery.” Thanks to the reflective property of the ellipse, a whispered message at one focus can be heard clearly by someone standing across the room at the other focus. If the elliptical ceiling has a major axis of 120 feet and a minor axis is 72 feet. How far apart are the foci?

Using the Foci

• How far apart are the foci of an ellipse with a major axis of 26 feet and a minor axis of 10 feet?

Using the Foci

• What is the standard form equation of the ellipse with foci at (±5, 0) and co-vertices at (0, ±8)?

• What is the standard form equation of the ellipse with foci at (0, ±√17) and co-vertices at (±6, 0)?