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Conic Sections

Explore the definition of an ellipse, its parts, and how to derive the formula. Learn about ellipses with different centers and how to find their equations, vertices, and foci. Access Geogebra examples for better comprehension.

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Conic Sections

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  1. Conic Sections The EllipsePart A

  2. Ellipse • Another conicsection formedby a plane intersecting acone • Ellipse formed when

  3. Definition of Ellipse • Set of all points in the plane … • Sum of distances from two fixed points(foci) is a positive constant View Geogebra Example

  4. Definition of Ellipse • Definition demonstrated by using two tacks and a length of string to draw an ellipse

  5. Graph of an Ellipse Note various parts of an ellipse

  6. Deriving the Formula • Note • Why? • Write withdist. formula • Simplify

  7. Deriving the Formula • Consider P at (0, b) • Isoscelestriangle • Legs = a • And a a

  8. Major Axis on y-Axis • Standard form of equation becomes • In both cases • Length of major axis = 2a • Length of minor axis = 2b Link to Animated Web Page

  9. Using the Equation • Given an ellipse with equation • Determine foci • Determine values for a,b, and c • Sketch the graph

  10. Find the Equation • Given that an ellipse … • Has its center at (0,0) • Has a minor axis of length 6 • Has foci at (0,4) and (0,-4) • What is the equation?

  11. Ellipses with Center at (h,k) • When major axis parallelto x-axis equation can be shown to be

  12. Ellipses with Center at (h,k) • When major axis parallelto y-axis equation can be shown to be

  13. Find Vertices, Foci • Given the following equations, find the vertices and foci of these ellipses centered at (h, k)

  14. Find the Equation • Consider an ellipse with • Center at (0,3) • Minor axis of length 4 • Focci at (0,0) and (0,6) • What is the equation?

  15. Assignment • Ellipses A • 1 – 43 Odd

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