Conic Sections

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

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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.

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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

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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

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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?

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Graphing An Ellipse On the TI

• Given equation of an ellipse
• We note that it is not a function
• Use this trick

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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)

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Graphing Ellipse in Geogebra

• Enter ellipse as quadratic in x and y

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Area of an Ellipse

• What might be the area of an ellipse?
• If the area of a circle ishow 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

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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 its messy)

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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