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Pre-Calc Ellipses Lesson 6.3

Pre-Calc Ellipses Lesson 6.3 If you take a circle---grab hold of a horizontal diameter At the ends and just stretch it out symmetrically, then The result is an ellipse with a horizontal major axis. Equation will be: x 2 + y 2 = 1

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Pre-Calc Ellipses Lesson 6.3

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  1. Pre-Calc Ellipses Lesson 6.3 If you take a circle---grab hold of a horizontal diameter At the ends and just stretch it out symmetrically, then The result is an ellipse with a horizontal major axis. Equation will be: x2 + y2 = 1 a2 b2 For a situation like this  we say it has a: i. horizontal major axis  which will be ‘2a’ units long; the coordinates of the endpoints of major axis will be (a,0) and (- a,0) (If the center is at the origin) ii. vertical minor axis  which will be ‘2b’ units long; the coordinates of the endpoints of minor axis will be (0,b) and (0, - b) (If the center is at the origin)

  2. Now, If you take that circle---grab hold of a veritcal diameter at each end and just stretch it out symmetrically, then the result is an ellipse with a vertical major axis. Equation will look like this: x2 + y2 = 1 b2 a2 For a situation like this  we say it has a: i. vertical major axis  which will be ‘2a’ units long; the coordinates of the endpoints of major axis will be (0,a) and (0,- a) (If the center is at the origin) ii. horizontal minor axis  which will be ‘2b’ units long; the coordinates of the endpoints of minor axis will be (b,0) and (- b,0) (If the center is at the origin) Now I know what you are thinking??

  3. How do you determine which is which? It all depends on the values located under the x2 and y2 whichever number is the greater is  a2 Now an ellipse has two other special points that are always located on the ‘major axis’  These points are called the ‘foci’ and the variable associated with these points is ‘c’. So depending on whether your major axis is horizontal or vertical is how you label the focus point! (c,0) & (-c,0) or (0,c) & (0,- c) Also the endpoints on your major axis  either (a,0) & (-a,0) or (0,a) & (0,- a) are called the: Vertices of the ellipse!!!!!! Now to help determine the values of ‘a’, ‘b’, and ‘c’

  4. We use a variation of an old friend: a2 - b2 = c2 Example 1: Find an equation of the ellipse with center at the origin, one vertex at (0,5) and one focus at (0,2). Sketch the ellipse and label the vertices, and the endpoints of the minor axis.

  5. Example 2: • Find an equation of an ellipse with center at the origin, • vertex at (4,0) and a minor axis that is 4 units long. • Find the coordinates of its ‘foci’.

  6. Example 3: Sketch the graphs of 4x2 + y2 = 64 and x + y = 4 on the same set of axes. Determine how many points of ‘solution’ there are to this system and ‘solve this system’ ALGEBRAICALLY!!!!!!

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