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Conic Sections

Conic Sections. Ellipse Part 3. Additional Ellipse Elements. Recall that the parabola had a directrix The ellipse has two directrices They are related to the eccentricity Distance from center to directrix = . Directrices of An Ellipse. An ellipse is the locus of points such that

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Conic Sections

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  1. Conic Sections EllipsePart 3

  2. Additional Ellipse Elements • Recall that the parabola had a directrix • The ellipse has two directrices • They are related to the eccentricity • Distance from center to directrix =

  3. Directrices of An Ellipse • An ellipse is the locus of points such that • The ratio of the distance to the nearer focus to … • The distance to the nearer directrix … • Equals a constant that is less than one. • This constant is the eccentricity.

  4. Directrices of An Ellipse • Find the directrices of the ellipse defined by

  5. Additional Ellipse Elements • The latus rectum is the distance across the ellipse at the focal point. • There is one at each focus. • They are shown in red

  6. Length = Latus Rectum • Consider the length of the latus rectum • Use the equation foran ellipse and solve for the y valuewhen x = c • Then double that distance

  7. Try It Out • Given the ellipse • What is the length of the latus rectum? • What are the lines that are the directrices?

  8. Graphing An Ellipse On the TI • Given equation of an ellipse • We note that it is not a function • Use this trick

  9. Graphing An Ellipse On the TI • Set Zoom Square • Note gaps dueto resolution • Graphing routine • Specify an x • Solve for zero of expression for y • Graph the (x,y)

  10. Graphing Ellipse in Geogebra • Enter ellipse as quadratic in x and y

  11. Area of an Ellipse • What might be the area of an ellipse? • If the area of a circle is…how might that relate to the area of the ellipse? • An ellipse is just a unit circle that has been stretched by a factor A in the x-direction, and a factor B in the y-direction

  12. Area of an Ellipse • Thus we could conclude that the area of an ellipse is • Try it with • Check with a definite integral (use your calculator … it’s messy)

  13. Assignment • Ellipses C • Exercises from handout 6.2 • Exercises 69 – 74, 77 – 79 • Also find areas of ellipse described in 73 and 79

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