Conic Sections The Ellipse Part A. Ellipse Another conic section formed by a plane intersecting a cone Ellipse formed when.

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Presentation transcript:

Conic Sections The Ellipse Part A

Ellipse Another conic section formed by a plane intersecting a cone Ellipse formed when

Definition of Ellipse Set of all points in the plane …  Sum of distances from two fixed points (foci) is a positive constant View Geogebra Example View Geogebra Example

Definition of Ellipse Definition demonstrated by using two tacks and a length of string to draw an ellipse

Graph of an Ellipse Note various parts of an ellipse

Deriving the Formula Note  Why? Write with dist. formula Simplify

Deriving the Formula Consider P at (0, b)  Isosceles triangle  Legs = a And a a

Major Axis on y-Axis Standard form of equation becomes In both cases  Length of major axis = 2a  Length of minor axis = 2b  Link to Animated Web Page Link to Animated Web Page

Using the Equation Given an ellipse with equation Determine foci Determine values for a, b, and c Sketch the graph

Find the Equation Given that an ellipse …  Has its center at (0,0)  Has a minor axis of length 6  Has foci at (0,4) and (0,-4) What is the equation?

Ellipses with Center at (h,k) When major axis parallel to x-axis equation can be shown to be

Ellipses with Center at (h,k) When major axis parallel to y-axis equation can be shown to be

Find Vertices, Foci Given the following equations, find the vertices and foci of these ellipses centered at (h, k)

Find the Equation Consider an ellipse with  Center at (0,3)  Minor axis of length 4  Focci at (0,0) and (0,6) What is the equation?

Assignment Ellipses A 1 – 43 Odd