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Sullivan Algebra and Trigonometry: Section 10.3 The Ellipse

Learn how to find the equation of an ellipse, graph ellipses, and discuss the properties of an ellipse. Examples and step-by-step explanations provided.

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Sullivan Algebra and Trigonometry: Section 10.3 The Ellipse

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  1. Sullivan Algebra and Trigonometry: Section 10.3The Ellipse • Objectives of this Section • Find the Equation of an Ellipse • Graph Ellipses • Discuss the Equation of an Ellipse • Work with Ellipses with Center at (h,k)

  2. Minor Axis Vertex Vertex F1: (-c,0) F2: (c,0) Major Axis An ellipse is the collection of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.

  3. Equation of an Ellipse: Center at (0,0); Foci at (c, 0) and (-c,0); Major Axis is Horizontal where a > b > 0 and b2 = a2 - c2 The major axis is the x - axis. The vertices are at (-a, 0) and (a, 0)

  4. Find the equation of an ellipse with center at the origin, one focus at (4, 0), and a vertex at (-5,0). Graph the equation. Since the given focus and vertex are on the x-axis, the major axis is the x-axis. The distance from the center to one of the foci is c = 4. The distance from the center to one of the vertices is a = 5. Use c and a to solve for b. b2 = a2 - c2 b2 = 52 - 42 = 25 - 16 = 9 b = 3

  5. (0,3) (-5,0) (5,0) F1: (-4,0) F2: (4,0) (0,-3) So, the equation of the ellipse is:

  6. Discuss the equation: Since the equation is written in the desirable form, a2 = 16 and b2 = 7 Since b2 = a2 - c2, it follows that c2 = a2 - b2 or c2 = 16 - 7 = 9. So, the foci are at (3,0) and (-3,0) The vertices are at (-4, 0) and (4, 0)

  7. Equation of an Ellipse: Center at (0,0); Foci at (0, c) and (0, -c); Major Axis is Vertical where a > b > 0 and b2 = a2 - c2 The major axis is the y - axis. The vertices are at (0, -a) and (0, a)

  8. Find the equation of an ellipse having one focus at (0, 2) and vertices at (0, 3) and (0, -3). Since the vertices lie on the y - axis, the major axis is vertical with a = 3. The distance from the focus to the center is c = 2. b2 = a2 - c2 b2 = 32 - 22 = 9 - 4 = 5 So, the equation of the ellipse is:

  9. If an ellipse with center at the origin and major axis coinciding with a coordinate axis is shifted horizontally h units and vertically k units, the resulting ellipse is centered at (h,k) and has the equation: Horizontal Major Axis Vertical Major Axis

  10. Find the equation of an ellipse with center at (2, -3), one focus at (3, -3), and one vertex at (5, -3). The center is at (h,k) = (2, -3). So h = 2 and k = -3 The center, focus, and vertex all lie on the line y = -3, so the major axis is parallel to the x-axis and the ellipse is horizontal. The distance from the center to the vertex is a = 3. The distance from the center to the focus is c = 1. To solve for b,

  11. b2 = a2 - c2 b2 = 32 - 12 = 9 - 1 = 8 So, the equation of the ellipse is:

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