1 Conic Sections Ellipse Part 3. 2 Additional Ellipse Elements Recall that the parabola had a directrix The ellipse has two directrices  They are related.

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Presentation transcript:

1 Conic Sections Ellipse Part 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 Latus Rectum Consider the length of the latus rectum Use the equation for an ellipse and solve for the y value when x = c  Then double that distance Length =

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  Must be graphed in two portions Solve for y

9 Graphing An Ellipse On the TI Use both results Set resolution to 1 to close gaps between upper and lower portion

10 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

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

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