1. 10.4 - Ellipses

2. ELLIPSE TERMS ca Minor axis Major axis EQUATION FORM Center at origin VERTICES CO-VERTICES MAJOR AXIS MAJOR length MINOR AXIS MINOR length FOCI a > b > 0 (0, ± b) horizontal (± a, 0) b Vertex Co-vertex 2a vertical (± c, 0) Focus 2b

3. ELLIPSE TERMS c a Minor axis Major axis EQUATION FORM Center at origin VERTICES CO-VERTICES MAJOR AXIS MAJOR length MINOR AXIS MINOR length FOCI a > b > 0 (± b, 0) vertical (0, ± a) b Vertex Co-vertex 2a horizontal (0, ± c ) Focus 2b

4. Write an equation in standard form of an ellipse that has a vertex at (0, –4), a co-vertex at (3, 0), and is centered at the origin. Writing an Equation of an Ellipse Since (0, –4) is a vertex of the ellipse, the other vertex is at (0, 4), and the major axis is vertical. Since (3, 0) is a co-vertex, the other co-vertex is at (–3, 0), and the minor axis is horizontal. So, a = 4, b = 3, a 2 = 16, and b 2 = 9. + = 1Standard form for an equation of an ellipse with a vertical major axis. x 2b2x 2b2 y 2a2y 2a2 + = 1Substitute 9 for b 2 and 16 for a 2. x 29x 29 y 2 16 An equation of the ellipse is + = 1. x 29x 29 y 2 16

5. Write an equation in standard form of an ellipse that has a vertex at (5, 0), a co-vertex at (0, 2), and is centered at the origin. Writing an Equation of an Ellipse

6. Write an equation in standard form of an ellipse Writing an Equation of an Ellipse

7. Graph and Label Find the foci and then graph

8. Graph and Label Find the foci and then graph

9. Find the Equation of an Ellipse Write an equation of the ellipse with foci at (0, ±4) and co- vertices at (±2, 0). Since the foci have coordinates (0, ±4), the major axis is vertical. Since c = 4 and b = 2, c 2 = 16, and b 2 = 4. c 2 = a 2 – b 2 Use the equation to find a 2. 16 = a 2 – 4Substitute 16 for c 2 and 4 for b 2. a 2 = 20Simplify. + = 1Substitute 20 for a 2 and 4 for b 2. x 2x 2 4 y 2y 2 20 An equation of the ellipse is + = 1. x 2x 2 4 y 2y 2 20

10. Find the Equation of an Ellipse Write an equation of the ellipse with foci at ( ±3,0) and co- vertices at (0, ± 1). Since the foci have coordinates ( ±3, 0), the major axis is horizontal. Since c = 3 and b = 1 c 2 = 9, and b 2 = 1. c 2 = a 2 – b 2 Use the equation to find a 2. 9 = a 2 – 1Substitute 9 for c 2 and 1 for b 2. a 2 = 10Simplify. + = 1Substitute 10 for a 2 and 1 for b 2. x 2x 2 10 y 2y 2 1 An equation of the ellipse is + = 1. x 2x 2 10 y 2y 2 1

11. Application of foci in ellipses  You may think that most objects in space that orbit something else move in circles, but that isn't the case. Although some objects follow circular orbits, most orbits are shaped more like "stretched out" circles or ovals. Mathematicians and astronomers call this oval shape an ellipse. The Sun isn't quite at the center of a planet's elliptical orbit. An ellipse has a point a little bit away from the center called the "focus". The Sun is at the focus of the ellipse. Because the Sun is at the focus, not the center, of the ellipse, the planet moves closer to and further away from the Sun every orbit

12. More on orbits… Orbits are ellipses. An ellipse can be like a circle, or it can be long and skinny. Mathematicians and astronomers use the term "eccentricity" to describe the shape of an orbit. Eccentricity = c/a. An orbit shaped almost like a circle has a low eccentricity close to zero. A long, skinny orbit has a high eccentricity, close to one http://windows2universe.org/physical_science/physics/mechanics/orbit/orbi t_shape_interactive.html http://windows2universe.org/physical_science/physics/mechanics/orbit/orbi t_shape_interactive.html