Conic Sections: The Ellipse

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Conic SectionsThe Ellipse
MAC 1140 Mrs. Kessler
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By changing the angle and location of
intersection, we can produce a circle, ellipse,
parabola or hyperbola or in the special case
when the plane touches the vertex a point, line
or 2 intersecting lines.
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Definition of an Ellipse

• An ellipse is the set of all points in a plane
  the sum of whose distances from two fixed points,
  is constant. These two fixed points are called
  the foci. The midpoint of the segment connecting
  the foci is the center of the ellipse.

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Apollonius of Perga, one of the greatest Greek
mathematicians of the time (circa 200 B.C.),
appears to have been the first to have
rigorously studied the conic sections. He
applied his work to his study of planetary motion
and used this to aid in the development of Greek
astronomy.
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On a far smaller scale, the electrons of an atom
move in an approximately elliptical orbit with
the nucleus at one focus.
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The ellipse has an important property that is
used in the reflection of light and sound waves.
Any light or signal that starts at one focus
will be reflected to the other focus. This
principle is used in lithotripsy, a medical
procedure for treating kidney stones. The
patient is placed in a elliptical tank of water,
with the kidney stone at one focus. High-energy
shock waves generated at the other focus are
concentrated on the stone, pulverizing it.
Statuary Hall in the U.S. Capital building is
elliptic. It was in this room that John Quincy
Adams, while a member of the House of
Representatives, discovered this acoustical
phenomenon. He situated his desk at a focal
point of the elliptical ceiling, easily
eavesdropping on the private conversations of
other House members located near the other focal
point.
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Standard Forms of the Equations of an Ellipse

• The standard form of the equation of an ellipse
  with center at the origin, and axes of lengths 2a
  and 2b
• If a gt b, the major axis is the x - axis.
• If b gt a, the major axis is the y - axis
• The vertices are on the major axis, a (or b)
  units from the center.

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Standard Forms of the Equations of an Ellipse
The foci are on the major axis, c units from the
center. c2 a2 b2 or c2 b2 a2
depending on whether a gt b or b gt a
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Example 1 find vertices, and foci
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Example 2
Graph and locate the foci 25x2 16y2 400.
Express the equation in standard form. Because
we want 1 on the right side, we divide both
sides by 400.
Since b gt a, the major axis is vertical.
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Example 2 cont.
Now find the endpoints of the horizontal minor
axis. Because a2 16, a 4. Thus, the
endpoints are (-4, 0) and (4, 0).
The foci are located at on the major axis, c
units away from the center. c2 25 16 9.
c 3.
Foci are at (0, -3) and (0, 3).
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Example 3

• Find the standard form of the equation of the
  ellipse centered at the origin with Foci (0,-3),
  (0,3) and vertices (0,-5), (0,5)

a 5 and c 3
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Standard Forms of Equations of Ellipses Centered
at (h,k)
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Example 4 Sketch, find vertices and foci
(2, 3)
Find center
Which is major axis?
How far from center are vertices?
How far from center are endpoints of the minor
axis?
How far from center are foci?
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Standard Forms of Equations of Ellipses Centered
at (h,k)
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Example 4 What is it?
4x2 9y2- 32x 36 y 64 0
4x2 - 32x 9y2 36 y -64
4(x2 - 8x ) 9(y2 4y ) -64
Complete the square!
4(x2 - 8x 16 ) 9(y2 4 y 4 ) -64
64 36
4(x - 4 )2 9(y2 2 )2 36