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10.4 Ellipses

10.4 Ellipses. Elliptical Orbits. Elliptical Galaxies. Elliptical Orbit Effect. Ellipses from Conic Sections. By Karen Kidwell. Review. What are the two other conic sections we have already discussed? Answer: Circles and Parabolas What is the equation for Circles? x 2 + y 2 = r 2

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10.4 Ellipses

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  1. 10.4 Ellipses Elliptical Orbits Elliptical Galaxies Elliptical Orbit Effect Ellipses from Conic Sections By Karen Kidwell

  2. Review. . . • What are the two other conic sections we have already discussed? • Answer: Circles and Parabolas • What is the equation for Circles? • x2 + y2 = r2 • What are the new equation for parabolas? • x2 = 4py or y2 = 4px Is that a parabola??

  3. Teach: Definition: • An ellipse is the set of all point P such that the sum of the distances between P and two distinct fixed points, called foci, is a constant. Interactive Demonstrations: http://www.mathopenref.com/constellipse1.html http://www.mathopenref.com/ellipse.html

  4. Vocabulary: • Vertices: The line through the foci and intersects the ellipse in two points • Major Axis: The line joining the two vertices • Center: Midpoint of the Major Axis • Co-vertices: The line perpendicular to the major axis at the center and intersects the ellipse in two points • Minor Axis: The line segment between the two co-vertices

  5. Diagram: b to b is the minor axis a to a is the major axis the c’s are the foci

  6. Equations • The standard form of the equation of an ellipse: Equation Major Axis Vertices Co-Vertices The foci of the ellipse are on the major axis, c units from the center where c2 = a2 – b2

  7. How does it work? • Draw the ellipse given by 9x2 + 16y2 = 144. • Steps: • First you must make the equation equal to 1. See previous slide. • Divide by 144 throughout the whole problem • Now you have x2/16 + y2/9 = 1 • What is a and b? (see equation!) • Answer a = 4 and b = 3 • So, along the horizontal (like x axis, notice it is below the x2), the length is 2a, 8 and along the vertical is 2b, 6, all centered around the origin (0, 0)

  8. Another problem: • Write an equation of the ellipse with the given characteristics and center at (0, 0). • Vertex: (-4, 0) and Focus: (2,0) • Steps: • We know a = what? and c = what? • Answer: a = 4 and c = 2 • How can we find b? • Answer: Use the formula c2 = a2 – b2 • What’s b? • 22 = 42 – b2; b2 = 12; b = 23 • So the equation is x2/16 + y2/12 = 1

  9. Practice Practice: • You try: • Write in standard form (if not already). Then, identify the vertices, co-vertices and foci of the ellipse: • 1. x2/25 + y2/16 = 1 • 2. 10x2 + 25y2 = 250 • 3. Graph the equation and identify the same above parts. x2/4 + y2/49 = 1 • Write the equation given • 4. Vertex: (0, -7) and Co-vertex: (-1, 0) • 5. Vertex: (15, 0) and Focus: (12, 0)

  10. Answers: • 1. vertices: (±5, 0), co-vertices: (0, ±4), and foci: (±3, 0) • 2. x2/25 + y2/10 = 1, vertices: (±5, 0), co-vertices: (0, ±10), and foci: (±15, 0) • 3. Graph of a vertical ellipse; vertices at (0,±7) and co-vertices at (±2, 0) • 4. x2/1 + y2/49 = 1 • 5. x2/225 + y2/81 = 1

  11. Apply: Apply • Both man-made objects, such as The Ellipse at the White House, and natural phenomena, such as the orbits of planets, involve ellipses.

  12. Solve: • 1. A portion of the White House lawn is called The Ellipse. It is 1060 feet long and 890 feet wide. • A. Write an equation of the Ellipse • B. The area of an ellipse is A= πab What is the area of The Ellipse at the White House?

  13. Solve: • 2. In its elliptical orbit, Mercury ranges from 46.04 million kilometers to 69.86 million kilometers from the center of the sun. The center of the sun is a focus of the orbit. Write an equation of the orbit.

  14. You are on your way to becoming an expert in: Ellipses

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