1 / 10

Math 143 Section 7.1 The Ellipse

Math 143 Section 7.1 The Ellipse. Ellipse. An ellipse is a set of points in a plane the sum of whose distances from two fixed points, called foci , is a constant. For any point P that is on the ellipse , d 2 + d 1 is always the same. P. d 2. d 1. F 1. F 2.

dieter-herring
Download Presentation

Math 143 Section 7.1 The Ellipse

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Math 143 Section 7.1The Ellipse

  2. Ellipse An ellipse is a set of points in a plane the sum of whose distances from two fixed points, called foci, is a constant. For any point P that is on the ellipse , d2 + d1 is always the same. P d2 d1 F1 F2

  3. Standard Form Equation of an Ellipse (x – h)2 (y – k)2 + = 1 a2 b2 The center of the ellipse is at the point (h, k) a is ½ the length of the horizontal axis b is ½ the length of the vertical axis Where c is the distance from the center to a focus point. c2 = a2 – b2 if a2 > b2 c2 = b2 – a2 if b2 > a2

  4. Graphing an Ellipse Graph: x2 y2 4 9 + = 1 Center: (0, 0) V Minor Axis: 4 (horizontal) F Major Axis: 6 (vertical) Vertices: (0, 3) and (0, - 3) F c2 = 9 – 4 = 5 V c = Ö5 = 2.24 Foci: (0, 2.24) and (0, -2.24)

  5. Graphing an Ellipse Graph: (x – 2)2 (y + 1)2 16 9 + = 1 Center: (2, -1) Major Axis: 8 (horizontal) Minor Axis: 6 (vertical) V Vertices: (6, -1) and (-2, -1) V c2 = 16 – 9 = 7 c = Ö 7 = 2.65 Foci: (4.65, -1) and (-0.65, -1)

  6. Finding an Equation of an Ellipse Find the equation of the ellipse given that Vertices are (0, 4), (0, -4) Foci are (0, 3), (0, -3) V Center: (0, 0) Major Axis: 8 (vertical) b = 4 and b2 = 16 Since c = 3, and c2 = b2 – a2 9 = 16 – a2 a2 = 7 V Equation: (x – h)2 (y – k)2 a2 b2 + = 1 x2 y2 7 9 + = 1

  7. Finding an Equation of an Ellipse Find the equation of the ellipse given the graph Then locate the foci of the ellipse Center: (-1, 1) V1 Major Axis: 6 (horizontal), so a = 3 Minor Axis: 2 (vertical), so b = 1 Equation: (x + 1)2 (y – 1)2 9 1 + = 1 c2 = a2 – b2 V2 c2 = 9 – 1 = 8 c = Ö8 = 2.83 (x – h)2 (y – k)2 a2 b2 + = 1 Foci are (1.83, 1), (-3.83, 1)

  8. Finding an Equation of an Ellipse Find the equation of the ellipse given that Foci are (-2, 0), (2, 0) y-intercepts: -3, 3 Major Axis must be horizontal since the foci are on the major axis Center: (0, 0) c= 2 and b = 3 F F c2 = a2 – b2 4 = a2 – 9 a2 = 13 Equation: (x – h)2 (y – k)2 a2 b2 + = 1 x2 y2 13 9 + = 1

  9. Converting an Equation Convert the following equation to standard form Then graph the ellipse and locate its foci 9x2 + 25y2 – 36x + 50y – 164 = 0 9(x2 – 4x + ___ ) + 25(y2 + 2y + ___) = 164 + ___ + ___ 4 1 36 25 9(x – 2)2 + 25(y + 1)2 = 225 (x – 2)2(y + 1)2 + = 1 25 9 c2 = 25 – 9 = 16 c = 4 F F Foci: (-2, -1) and (6, -1)

  10. Application Problems A semielliptical archway has a height of 20 feet at its midpoint and a width of 50 feet. Can a truck that is 14 ft high and 10 ft wide drive under the archway without moving into the oncoming lane? P The real question is “What is the value of y at point P when x = 10” ? 20 10 Equation of the ellipse: x2 y2 50 + = 1 625 400 16x2 + 25y2 = 10,000 Yes, the truck will be able to drive under the archway without moving into the oncoming lane. When x = 10, 1600 + 25y2 = 10,000 25y2 = 8400 y2 = 336 y = 18.3

More Related