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Ellipses

Ellipses. Topic 11.3. Definitions. Ellipse: set of all points where the sum of the distances from the foci is constant Major Axis: axis on which the foci lie; the longer axis of symmetry Minor Axis: the shorter axis of symmetry. Two Standard Equations. Horizontal Ellipse: Foci:

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Ellipses

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  1. Ellipses Topic 11.3

  2. Definitions • Ellipse: set of all points where the sum of the distances from the foci is constant • Major Axis: axis on which the foci lie; the longer axis of symmetry • Minor Axis: the shorter axis of symmetry

  3. Two Standard Equations • Horizontal Ellipse: • Foci: • Vertical Ellipse: • Foci:

  4. Writing in Standard Form • Complete the square for both the x-terms and y-terms and move the constant to the other side of the equation • Divide all terms by the constant

  5. Example: Group terms Complete the square Simplify each group Divide by constant

  6. You Try!:x2 + 4y2 - 8x - 48y + 124 = 0

  7. You Try!:4x2 + 9y2 - 40x + 36y + 100 = 0

  8. Graphing the ellipse • Put equation in standard form • Graph the center (h, k) • Graph the foci (look at the equation to determine your direction) • Graph a units and –a units from the center to get the end points of major (horizontally if under x, vertically if under y) • Graph b units and –b units from the center to get the end points of minor (vertically if under x, horizontally if under y) • Connect the end points!

  9. Example: 1) Graph Center 2) Graph Foci . 3) Graph Endpoints of both axis . 4) Graph Ellipse

  10. You Try!: (x – 4)2 + (y – 1)2 = 1 64 16

  11. Answer the following for the last problem • a = b = • 1. horizontal or vertical • 2. center/shift • 3. vertices • 4. length major axis • 5. length minor axis • 6. foci

  12. You Try!Write the following equation in standard form, then graph it.

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