SEC 11'3The Ellipse

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SEC 11.3 The Ellipse
Definition An ellipse is the set of all points
in the plane, the sum of whose distances from two
fixed points (foci) is a constant.
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Standard Forms of the Equation of an Ellipse

• (a) Center (0, 0)
• (b) Vertices (- a, 0) and (a, 0)
• (c) Co-vertices (0, - b) and (0, b)
• (d) Foci (- c, 0) and (c, 0), where
• (e) The major axis is on the x-axis, and has
  length 2a.
• (f) The length of the minor axis is 2b.

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Standard Forms of the Equation of an Ellipse

• (a) Center (0, 0)
• (b) Vertices (0, -a) and (0, a)
• (c) Co-vertices (-b, 0) and (b, 0)
• (d) Foci (0, -c) and (0, c), where
• (e) The major axis is on the y-axis, and has
  length 2a.
• (f) The length of the minor axis is 2b.

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Standard Forms of the Equation of an Ellipse

• (a) Center (h, k)
• (b) Vertices ( h - a, k) and ( h a, k)
• (c) Co-vertices (h, k - b) and ( h, k b)
• (d) Foci ( h - c, k) and ( h c, k), where
  
• (e) The major axis is parallel to the x-axis,
  and has
• length 2a.
• (f) The length of the minor axis is 2b.

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Standard Forms of the Equation of an Ellipse

• (a) Center (h, k)
• (b) Vertices ( h, k - a) and ( h, k a)
• (c) Co-vertices ( h - b, k) and ( h b, k)
• (d) Foci ( h, k - c) and ( h, k c), where
• (e) The major axis is parallel to the y-axis,
  and has length 2a.
• (f) The length of the minor axis is 2b.

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Eccentricity of an Ellipse

• Definition
• The eccentricity (denoted by e) of an ellipse
  measures how
• far an ellipse is from a circle.
• where c is the distance from the center to a
  focus and a is
• one-half the length of the major axis.
• 0 lt e lt 1.

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SEC 11.4 Hyperbolas
Definition A hyperbola is the set of all points
in the plane, the difference of whose distances
from two fixed points (foci) is a constant.
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Standard Forms of the Equation of a Hyperbola

• (a) Center (0, 0)
• (b) Vertices (-a, 0) and (a, 0)
• (c) Foci (-c, 0) and (c, 0), where
• (d) The transverse axis is the x-axis
• (e) The conjugate axis is the y-axis
• (f) The asymptotes are

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Standard Forms of the Equation of a Hyperbola

• (a) Center (0, 0)
• (b) Vertices (0, -a) and (0, a)
• (c) Foci (0, -c) and (0, c), where
• (d) The transverse axis is the y-axis
• (e) The conjugate axis is the x-axis
• (f) The asymptotes are

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Standard Forms of the Equation of a Hyperbola

• (a) Center (h, k)
• (b) Vertices (h - a, k) and (h a, k)
• (c) Foci (h -c, k) and (h c, k), where
• (d) The transverse axis is parallel to the
  x-axis
• (e) The conjugate axis is parallel to the
  y-axis
• (f) The asymptotes are

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Standard Forms of the Equation of a Hyperbola

• (a) Center (h, k)
• (b) Vertices (h, k - a) and (h, k a)
• (c) Foci (h, k -c) and (h, k c), where
• (d) The transverse axis is parallel to the
  y-axis
• (e) The conjugate axis is parallel to the
  x-axis
• (f) The asymptotes are

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Eccentricity (e) of a Hyperbola

• Definition
• where c is the distance from the center to a
  focus and a is
• one-half the length of the transverse axis.
• e gt 1.

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A FLOW CHART for Conic Sections