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The Ellipse

Unit #4 Conics. The Ellipse. The Ellipse. An ellipse is the set of all points in a plane whose distances from two fixed points in the plane, the foci, is constant . The line through the foci is called the focal or major axis. Minor Axis. Major Axis. Focus 1. Focus 2. Point.

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The Ellipse

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  1. Unit #4 Conics The Ellipse

  2. The Ellipse An ellipse is the set of all points in a plane whose distances from two fixed points in the plane, the foci, is constant. The line through the foci is called the focal or major axis Minor Axis Major Axis Focus 1 Focus 2 Point PF1 + PF2 = constant

  3. When the circle is wider than it is taller this is called a horizontal shape.

  4. When the circle is taller than it is wider is called a vertical shape.

  5. The Standard Forms of the Equation of the Ellipse [cont’d] The standard form of an ellipse centred at any point (h, k) with the major axis of length 2a parallel to the x-axis and a minor axis of length 2b parallel to the y-axis, is: (h, k) This is called a what kind of shape?

  6. The Standard Forms of the Equation of the Ellipse [cont’d] The standard form of an ellipse centred at any point (h, k) with the major axis of length 2a parallel to the y-axis and a minor axis of length 2b parallel to the x-axis, is: (h, k) This is called a what kind of shape?

  7. The Pythagorean Property b a a2 = b2 + c2 b2 = a2 -c2 c2 = a2 -b2 c F2(c, 0) F1(-c, 0) Length of major axis: 2a Length of minor axis: 2b Vertices: (a, 0) and (-a, 0) Foci: (-c, 0) and (c, 0)

  8. Finding the General Form of the Ellipse The general form of the ellipse 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

  9. Finding the Center, Vertices, and Foci State the coordinates of the vertices, the coordinates of the foci, and the lengths of the major and minor axes of the ellipse, defined by each equation. a) The center of the ellipse is (0, 0). Since the larger number occurs under the x2, the major axis lies on the x-axis. a a2=16; a=4; Since the major axis is 2a, it equals 8. b b2=9; b =3; so the length of the minor axis is 2b = 6. c The coordinates of the vertices are (4, 0) and (-4, 0). To find the coordinates of the foci, use the Pythagorean property: c2 = a2 - b2 = 42 - 32 = 16 - 9 = 7 The coordinates of the foci are: and

  10. You Do: Finding the Center, Vertices, and Foci b) 4x2 + 9y2 = 36 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 6. a b The length of the minor axis is 4. c The coordinates of the vertices are (3, 0) and (-3, 0). To find the coordinates of the foci, use the Pythagorean property. c2 = a2 - b2 = 32 - 22 = 9 - 4 = 5 The coordinates of the foci are: and

  11. Finding the Equation of the Ellipse With Centre at (0, 0) a) Find the equation of the ellipse with centre at (0, 0), foci at (5, 0) and (-5, 0), a major axis of length 16 units, and a minor axis of length 8 units. Since the foci are on the x-axis, the major axis is the x-axis. The length of the major axis is 16 so a = 8. The length of the minor axis is 8 so b = 4. Standard form 64 64 x2 + 4y2 = 64 x2 + 4y2 - 64 = 0 General form

  12. Finding the Equation of the Ellipse b) Centre at (0, 0) The length of the major axis is 12 on the y-axis so a = 6. The length of the minor axis on the x-axis is 6 so b = 3. 36 36 4x2 + y2 = 36 4x2 + y2 - 36 = 0 General form Standard form

  13. 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

  14. Convert from General form to Standard Form Example)x2 + 4y2 - 2x + 8y - 11 = 0 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 Since the larger number occurs under the x2, the major axis is parallel to the x-axis. h = k = a = b = 1 -1 4 2 c2 = a2 - b2 = 42 - 22 = 16 - 4 = 12 The centre is at (1, -1). The major axis, parallel to the x-axis, has a length of 8 units. The minor axis, parallel to the y-axis, has a length of 4 units. The foci are at and

  15. Sketching the Graph of the Ellipse [cont’d] x2 + 4y2 - 2x + 8y - 11 = 0 Centre (1, -1) (1, -1) F2 F1

  16. 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 6 4 c2 = a2 - b2 = 62 - 42 = 36 - 16 = 20 The centre is at (1, -5). The major axis, parallel to the y-axis, has a length of 12 units. The minor axis, parallel to the x-axis, has a length of 8 units. The foci are at: and

  17. Sketching the Graph of the Ellipse [cont’d] 9x2 + 4y2 - 18x + 40y - 35 = 0 F1 F2

  18. General Effects of the Parameters A and C Ax2 + Cy2 + Dx + Ey + F = 0 When A ≠ C, and AxC > 0, the resulting conic is an ellipse. If | A | > | C |, it is a vertical ellipse. If | A | < | C |, it is a horizontal ellipse. The closer in value A is to C, the closer the ellipse is to a circle. This distance is called the Eccentricity.

  19. Definition of Eccentricity of an Ellipse The eccentricity of an ellipse is

  20. Assignment Pages 652-653 A 2, 4, 6, 7-10, 16, 22, 24, 26, 28, 30, 34, 38, 40, 45-48, 50

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