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Conic Sections

EllipsePart 3

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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 =

2a a

e c

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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.

cea

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Directrices of An Ellipse

• Find the directrices of the ellipse defined by

2 2

149 35

x y

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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

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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

Length =

22b

a

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Try It Out

• Given the ellipse

• What is the length of the latus rectum?

• What are the lines that are the directrices?

2 23 2

116 9

x y

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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

2 23 2

125 36

x y

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Graphing An Ellipse On the TI

• Use both results

Set resolution to 1 to close gaps between

upper and lower portion

Set resolution to 1 to close gaps between

upper and lower portion

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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

2r

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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)

a b 2 2

136 25

x y

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Assignment

• Ellipses C

• Exercises from handout 6.2

• Exercises 69 – 74, 77 – 79

• Also find areas of ellipse described in 73 and 79