10 – 4 Ellipses. Ellipse Center (0, 0) Writing an Equation What is an equation in standard form of an ellipse centered at the origin with vertex.

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Presentation transcript:

10 – 4 Ellipses

Ellipse

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 c 2 = a 2 – b 2 to find the foci of ±c Vertical ellipse: (0, ±c) Horizontal ellipse (±c, 0) What are the foci of the ellipse with the equation 25x 2 + 9y 2 = 225 What are the foci of the ellipse with the equation 36x y 2 = 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)?