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The Ellipse

Chapter 3 Conics. 3.4. The Ellipse. 3.4. 1. MATHPOWER TM 12, WESTERN EDITION. The Ellipse. An ellipse is the conic produced by stretching a circle along the vertical or horizontal axis of symmetry or both. The stretches of

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The Ellipse

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  1. Chapter 3 Conics 3.4 The Ellipse 3.4.1 MATHPOWERTM 12, WESTERN EDITION

  2. The Ellipse An ellipse is the conic produced by stretching a circle along the vertical or horizontal axis of symmetry or both. The stretches of the ellipse are related to the length of the minor axis and the major axis. In fact, the stretches are equivalent to half the length of each axis. Major Axis is the longer axis. Minor axis is shorter axis. Vertex Minor Axis Major Axis Vertex Vertex Vertex Vertex 3.4.2

  3. The Standard Forms of the Equation of the Ellipse The standard form of an ellipse centered at the origin with the major axis of length 2a along the x-axis and a minor axis of length 2b along the y-axis, is: The horizontal stretch is the value of a in the equation and the vertical stretch is the value of b. 3.4.3

  4. The Standard Forms of the Equation of the Ellipse [cont’d] The standard form of an ellipse centred at the origin with the major axis of length 2b along the y-axis and a minor axis of length 2a along the x-axis, is: Important Note: The value of a is always the horizontal stretch and the value of b is always the vertical stretch in this standard form equation. 3.4.4

  5. The Standard Forms of the Equation of the Ellipse [cont’d] b a (h, k) The standard form of an ellipse centered at any point (h, k) with the major axis parallel to the x-axis with by a horizontal stretch of a and a minor axis parallel to the y-axis with a vertical of b, is: Note: horizontal translation of h units to get to the center. Note: vertical translation of k units to get to the center. 3.4.5

  6. The Standard Forms of the Equation of the Ellipse [cont’d] b a (h, k) The standard form of an ellipse centred at any point (h, k) with the major axis parallel to the y-axis and a minor axis parallel to the x-axis, is: 3.4.7

  7. Finding the General Form of the Ellipse The general form of the ellipse (and every other conic) is: Ax2 + Cy2 + Dx + Ey + F = 0 AxC > 0 and A ≠ C The general form may be found by expanding the standard form and then simplifying: [ ] 225 25x2 + 9y2 - 200x + 36y + 211 = 0 3.4.7

  8. Finding the Centre, Axes, and Vertices State the coordinates of the vertices,and the lengths of the major and minor axes, the vertices and the domain and range of the ellipse defined by each equation. a) The centre of the ellipse is (0, 0). Since the larger number occurs under the x2, the major axis lies on the x-axis. The length of the major axis is 8. b a The length of the minor axis is 6. The coordinates of the vertices are (4, 0), (-4, 0), (0, -3) and (0, 3) . The domain is The range is 3.4.8

  9. Finding the Equation of the Ellipse With Centre at (0, 0) a) Find the equation of the ellipse with centre at (0, 0), a vertical major axis of length 16 units, and a horizontal minor axis of length 8 units. The length of the vertical major axis is 16 so b = 8. The length of the horizontal minor axis is 8 so a = 4. Standard form 64 64 4x2 + y2 = 64 4x2 + y2 - 64 = 0 General form 3.4.9

  10. Finding the Equation of the Ellipse With Centre at (h, k) • Find the equation for the ellipse with the centre at (3, 2), • passing through the points (8, 2), (-2, 2), (3, -5), and (3, 9). The major axis is parallel to the y-axis and has a length of 14 units, so a = 5. The minor axis is parallel to the x-axis and has a length of 10 units, so b = 7. The centre is at (3, 2), so h = 3 and k = 2. (3, 2) Standard form 49(x - 3)2 + 25(y - 2)2 = 1225 49(x2 - 6x + 9) + 25(y2 - 4y + 4) = 1225 49x2 - 294x + 441 + 25y2 - 100y + 100 = 1225 49x2 + 25y2 -294x - 100y + 541 = 1225 49x2 + 25y2 -294x - 100y - 684 = 0 General form 3.4.10

  11. Finding the Equation of the Ellipse With Centre at (h, k) b) The major axis is parallel to the x-axis and has a length of 12 units, so a = 6. The minor axis is parallel to the y-axis and has a length of 6 units, so b = 3. The centre is at (-3, 2), so h = -3 and k = 2. (-3, 2) Standard form (x + 3)2 + 4(y - 2)2 = 36 (x2 + 6x + 9) + 4(y2 - 4y + 4) = 36 x2 + 6x + 9 + 4y2 - 16y + 16 = 36 x2 + 4y2 + 6x - 16y + 25 = 36 x2 + 4y2 + 6x - 16y - 11 = 0 General form 3.4.11

  12. Analysis of the Ellipse “Completing the Square” • Given x2 + 4y2 - 2x + 8y - 11 = 0, convert to standard form and • determine the center and the domain and range. x2 + 4y2 - 2x + 8y - 11 = 0 (x2 - 2x ) + (4y2 + 8y) - 11 = 0 (x2 - 2x + _____) + 4(y2 + 2y + _____) = 11 + _____ + _____ 1 4 1 1 (x - 1)2 + 4(y + 1)2 = 16 1 -1 4 2 h = k = a = b = Since the larger number occurs under the x2, the major axis is parallel to the x-axis. The centre is at (1, -1). The domain is given by Center – stretch Center + stretch 1 + 4 or 1 – 4 The range works the same way using y-components. 3.4.12

  13. Sketching the Graph of the Ellipse [cont’d] x2 + 4y2 - 2x + 8y - 11 = 0 Centre (1, -1) (1, -1) 3.4.13

  14. Analysis of the Ellipse b)9x2 + 4y2 - 18x + 40y - 35 = 0 9x2 + 4y2 - 18x + 40y - 35 = 0 (9x2 - 18x ) + (4y2 + 40y) - 35 = 0 9(x2 - 2x + _____) + 4(y2 + 10y + _____) = 35 + _____ + _____ 1 25 9 100 9(x - 1)2 + 4(y + 5)2 = 144 Since the larger number occurs under the y2, the major axis is parallel to the y-axis. h = k = a = b = 1 -5 4 6 The centre is at (1, -5). 3.4.14

  15. Sketching the Graph of the Ellipse [cont’d] 9x2 + 4y2 - 18x + 40y - 35 = 0 3.4.15

  16. Graphing an Ellipse Using a Graphing Calculator (x - 1)2 + 4(y + 1)2 = 16 4(y + 1)2 = 16 - (x - 1)2 3.4.16

  17. General Effects of the Parameters A and C When A ≠ C in the general form of the ellipse equation, Ax2 + Cy2 + Dx + Ey + F = 0 and AxC > 0, the resulting conic is an ellipse. Note: If A = C then the conic is a circle in every case. 3.4.17

  18. Practice: • 1. Given the equation in general form determine the conic. • 3x2 + 3y2 + 8x + 6y – 28 = 0 • 12x2 + y2 + 3x – 12 = 0 2. Given the following equation, determine center, domain, range and intercepts. 3. Convert to standard form. x2 + 4y2 - 2x + 8y - 11 = 0 3.4.18

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