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Ellipses

Ellipses. Objectives: Write the standard equation for an ellipse given sufficient information Given an equation of an ellipse, graph it and label the center, vertices, co-vertices, and foci. P. F 1. F 2. F 1 P + F 2 P = 2a . Definition of Ellipse.

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Ellipses

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  1. Ellipses Objectives: Write the standard equation for an ellipse given sufficient information Given an equation of an ellipse, graph it and label the center, vertices, co-vertices, and foci

  2. P F1 F2 F1P + F2P = 2a Definition of Ellipse An ellipse is the set of all points P in a plane such that the sum of the distances from P to two fixed points, F1 and F2, called the foci, is a constant.

  3. y x2 y2 + = 1 (0, b) V2 (a, 0) V1(–a, 0) a2 b2 x O F1(–c, 0) F2 (c, 0) (0, –b) Standard Equation of an Ellipse Horizontal Major Axis: a2 > b2 a2 – b2 = c2 length of major axis: 2a length of minor axis: 2b

  4. y x2 y2 + = 1 V2 (0, a) b2 a2 F2 (0, c) (–b, 0) (b, 0) x O F1(0, –c) V1(0, –a) Standard Equation of an Ellipse Vertical Major Axis: a2 > b2 a2 – b2 = c2 length of major axis: 2a length of minor axis: 2b

  5. 8 6 x2 y2 4 + = 1 100 36 2 -6 4 6 8 -2 -8 -4 2 -2 -4 -6 -8 Example 1 Write the standard equation for an ellipse with foci at (-8,0) and (8,0) and with a major axis of 20. Sketch the graph. length of major axis: 2a 2a = 20, so a = 10 a2 – b2 = c2 102 – b2 = 82 b2 = 100 - 64 b2 = 36, so b = 6

  6. x2 y2 + = 1 16 49 Example 2 Find the vertices and co-vertices of the ellipse. vertices: (0,7) and (0,-7) co-vertices: (4,0) and (-4,0)

  7. 8 6 x2 y2 4 + = 1 16 64 2 -6 4 6 8 -2 -8 -4 2 -2 -4 -6 -8 Example 3 Write the standard equation of the ellipse. length of major axis: 2a 2a = 16, so a = 8 length of minor axis: 2b 2b = 8, so b = 4

  8. Practice Write the standard equation for an ellipse with foci at (5,0) and (-5,0) and with vertices at (9,0) and (-9,0). Sketch the graph.

  9. (x – h)2 (y – k)2 + = 1 a2 b2 Standard Equation of a Translated Ellipse Horizontal Major Axis: a2 > b2 a2 – b2 = c2 length of major axis: 2alength of minor axis: 2b

  10. (x – h)2 (y – k)2 + = 1 b2 a2 Standard Equation of a Translated Ellipse Vertical Major Axis: a2 > b2 a2 – b2 = c2 length of major axis: 2alength of minor axis: 2b

  11. Example 1 An ellipse is defined by the equation 4x2 + 9y2 – 16x + 18y = 11. Write the standard equation and identify the coordinates of the center, vertices, co-vertices, and foci. Sketch the graph of the ellipse. 4x2 – 16x + 9y2 + 18y = 11 4(x2 – 4x) + 9(y2 + 2y) = 11 4(x2 – 4x + 4) + 9(y2 + 2y + 1) = 11 + 4(4) + 9(1) 4(x – 2)2 + 9(y + 1)2 = 36

  12. 6 4 2 6 -2 -6 -4 2 4 -2 -4 -6 Example 1 An ellipse is defined by the equation 4x2 + 9y2 – 16x + 18y = 11. Write the standard equation and identify the coordinates of the center, vertices, co-vertices, and foci. Sketch the graph of the ellipse. center: (2,-1) a2 = 9, so a = 3 vertices: (-1,-1) and (5,-1) b2 = 4, so b = 2 co-vertices: (2,1) and (2,-3) a2 – b2 = c2 9 - 4 = c2

  13. Practice Write the standard equation for the ellipse 9x2 + 16y2 – 36x – 64y – 44 = 0. Identify the center, vertices, co-vertices, and foci.

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