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Section 8.3 Ellipses

Circle Ellipse. Parabola Hyperbola. Section 8.3 Ellipses. Ellipse:. Besides having the two foci, an ellipse also has a major and minor axis, vertices at the end of the major axis and center point where the two axes cross. Standard Equations for an Ellipse

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Section 8.3 Ellipses

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  1. Circle Ellipse Parabola Hyperbola Section 8.3 Ellipses

  2. Ellipse:

  3. Besides having the two foci, an ellipse also has a major and minor axis, vertices at the end of the major axis and center point where the two axes cross.

  4. Standard Equations for an Ellipse Major axis Parallel to x - axis x 2 y 2 a 2 b 2 + = 1 (0, b) (a,0) b V F F a V (- a, 0) (- c , 0) (c, 0) (0, - b) Center = (0, 0) Vertices (a, 0), (- a, 0) Minor Intercepts (0, b), (0, -b) Foci (c, 0), (- c, 0) c2 = a2 - b2 a > b > 0 Major Axis = 2a Minor Axis = 2b (0, 0)

  5. Standard Equations for an Ellipse Major axis parallel to y - axis V (0,a) (0,c) x 2 y 2 b 2 a 2 F + = 1 b (b,0) (-b,0) a (0,-c) F V (0,- a,) a > b > 0 Center = (0, 0) Vertices (0, a), (0, - a) (0, 0) Minor Intercepts (b, 0), (- b, 0) Major Axis = 2a Minor Axis = 2b c2 = a2 - b2 Foci (0, c), (0, - c)

  6. Ellipse Sketch, Find Foci, Length of Minor and Major Axis For Center at the origin. x2 y2 16 9 3 + = 1 - 4 7 - 7 | | | | | | | | | | | | | | | | | | | | | | | 4 c2 = a2 - b2 = 16 - 9 = 7 c =  7 - 3 b2 = 9 b = 3 a2 = 16 a = 4 Vertices = (4, 0) & (- 4, 0) Minor intercepts = (0, 3) & (0,- 3) Foci = (7, 0) & (- 7, 0) Maj. Axis=2·a=2(4)=8Min. Axis=2·b=2(3)=6

  7. Ellipse Sketch, Find Foci, Length of Minor and Major Axis For Center at the origin. x2 y2 16 81 9 65 + = 1 - 4 | | | | | | | | | | | | | | | | | | | | | | | 4 c2 = a2 - b2 = 81 - 16 = 65 c =  65 - 65 - 9 a2 = 81 a = 9 b2 = 16 b = 4 Vertices = (0, 9) & (0, - 9) Minor intercepts = (4,0) & (- 4,0) Foci = (0,  65) & (0, -  65) Maj. Axis=2·a=2(9)=18Min. Axis=2·b=2(4)=8

  8. Graph the Ellipse Needs to be set equal to 1. Vertices: (0,-4) and (0,4) Minor Intercepts: (-1,0) and (1,0)

  9. Find the equation of the ellipse Foci: (-1,0) and (1,0) Vertices: (-3,0) and (3,0) Therefore a = 3 and c = 1

  10. Ellipse Find an equation of an ellipse in the form x2 y2 a2 b2 + = 1 x2 y2 256 225 + = 1 1. When Major axis is on x-axis Major axis length = 32 Minor axis length = 30 Therefore, a = 32 ÷ 2 = 16 b = 30 ÷ 2 = 15 b2 = 225 a2 = 256

  11. Ellipse Find an equation of an ellipse in the form x2 y2 b2 a2 + = 1 x2 y2 15 64 + = 1 2. Major axis on y-axis Major axis length = 16 Distance from Foci to Center = 7 Therefore, c = 7 a = 16 ÷ 2 = 8 a2 = 64 c2 = a2 – b2 b2 = a2 – c2 = 64 – 49 = 15

  12. x2 y2 100 36 x2 y2 a2 b2 + = 1 + = 1 | | | | | | | | | | | | | | | | | | | | | | | a = 10 b = 6 Find the equation of the ellipse in the form below if thee center is the origin. a2 = 100 b2 = 36

  13. Translations Ellipses translate just like circles and parabolas do…by using h and k in the standard equation. This is for a horizontal major axis, switch a and b for a vertical major axis…if your equation isn’t in this form you will need to complete the square to make it so…

  14. Graph the ellipse Center: (-1,3) Major axis parallel to x-axis Place a point 3 units right and left of center Place a point 1 unit above and below the center. Foci are about 2.8 units to the left and right of center.

  15. Graph the ellipse

  16. Major axis is parallel to the y-axis Center is (-4,1) Place 2 points 1.4 unit right and left of center Place 2 points 2.8 units up and down from center

  17. Write the equation of the ellipse Foci: (2,-2) and (4,-2) Vertices: (0,-2) and (6,-2) Plug into standard form: Center is halfway between the vertices so the point (3,-2) We know a = 3 and c = 1

  18. Write the equation of the ellipse Major axis vertical with length of 6 and minor axis length of 4 centered at (1,-4)

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