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 1. Complete the square for both the x- terms and y-terms and move the constant to the other side of the equation 2. Divide all terms by the constant

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

6. You Try!: x 2 + 4y 2 - 8x - 48y + 124 = 0

7. You Try!: 4x 2 + 9y 2 - 40x + 36y + 100 = 0

8. Graphing the ellipse 1. Put equation in standard form 2. Graph the center (h, k) 3. Graph the foci (look at the equation to determine your direction) 4. Graph a units and –a units from the center to get the end points of major (horizontally if under x, vertically if under y) 5. Graph b units and –b units from the center to get the end points of minor (vertically if under x, horizontally if under y) 6. 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.