ellipse

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ellipse
By Jeffrey Bivin Lake Zurich High
School jeff.bivin_at_lz95.org
Last Updated March 11, 2008
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Ellipse

• The set of all co-planar points whose sum of the
  distances from two fixed points (foci) are
  constant.

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Ellipse

• The set of all points whose sum of the distances
  from two fixed points (foci) are constant.

Look at an animation
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Ellipse

• Definition of major and minor axis

minor axis
major axis
5
Ellipse
major axis
minor axis
6
Ellipse

• Definition of vertices
• and co-vertices

co-verticies
verticies
verticies
endpoint ofmajor axis
co-verticies
endpoint ofminor axis
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Ellipse
Distance from center to vertex a
Distance from center to co-vertex b
Distance from center to foci c
a
b
c
a
Length of major axis 2a
Length of minor axis 2b
a2 b2 c2
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Ellipse
a
b
c
a
Distance from center to vertex a
a2 b2 c2
Distance from center to co-vertex b
Distance from center to foci c
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Ellipse
a
a
c
a2 b2 c2
b
Distance from center to vertex a
Distance from center to co-vertex b
Distance from center to foci c
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Ellipse
The Latus Rectum (LR) is a chord passing through
the focus that is perpendicular to the major axis.
The length of the L.R. is
Distance from center to vertex a
Distance from center to co-vertex b
Distance from center to foci c
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Graph the following Ellipse
Center (2, -3)
a 5 in x direction
b 3 in y direction
(2, 0)
3
(2, -3)
(-3, -3)
(7, -3)
5
5
3
(2, -6)
12
Graph the following Ellipse
a 5 b 3
a2 b2 c2
52 32 c2
25 9 c2
foci
16 c2
4 c
(2, 0)
3
(-2, -3)
(6, -3)
(2, -3)
(-3, -3)
(7, -3)
5
5
3
(2, -6)
13
Graph the following Ellipse
a 5 b 3 c 4
Find the Length of the LR.
(2, 0)
3
(2, -3)
(-3, -3)
(7, -3)
5
5
3
(2, -6)
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Graph the following Ellipse
Center (2, -3)
Major Axis length 10 endpoints (-3, -3)
(7, -3)
Minor Axis length 6 endpoints (2, 0)
(2, -6)
Foci (-2, -3) (6, -3)
(2, 0)
Length of LR
3
(-2, -3)
(6, -3)
(2, -3)
(-3, -3)
(7, -3)
5
5
3
(2, -6)
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Graph the following Ellipse
x 1 0
y - 5 0
x -1
y 5
(-1, 12)
Center (-1, 5)
7
a 7 in y direction
(-1, 5)
(-5, 5)
(3, 5)
4
4
b 4 in x direction
7
(-1, -2)
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Graph the following Ellipse
a 7 b 4
a2 b2 c2
72 42 c2
49 16 c2
(-1, 12)
33 c2
foci
7
(-1, 5)
(-5, 5)
(3, 5)
4
4
7
(-1, -2)
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Graph the following Ellipse
Find theLength of the LR.
a 7 b 4 c
(-1, 12)
7
(-1, 5)
(-5, 5)
(3, 5)
4
4
7
(-1, -2)
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Graph the following Ellipse
Center (-1, 5)
Major Axis length 14 endpoints (-1,
-2) (1, 12)
(-1, 12)
Minor Axis length 8 endpoints (-5,
5) (3, 5)
7
(-1, 5)
(-5, 5)
(3, 5)
4
4
Foci
7
Length of LR
(-1, -2)
19
Graph the following Ellipse
4x2 8x 9y2 54y 52 3
(4x2 8x ) (9y2 54y ) 3 - 52
4(x2 2x 12) 9(y2 6y 32) -49 4 81
4(x 1)2 9(y 3)2 36
36
9
4
20
Graph the following Ellipse
Center (-1, -3)
a 3 in x direction
b 2 in y direction
(-1, -1)
2
(-1, -3)
(-4, -3)
(2, -3)
3
3
2
(-1, -5)
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Graph the following Ellipse
a 3 b 2
a2 b2 c2
32 22 c2
9 4 c2
foci
5 c2
(-1, -1)
2
(-1, -3)
(-4, -3)
(2, -3)
3
3
2
(-1, -5)
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Graph the following Ellipse
a 3 b 2 c
Find the Length of the LR.
(-1, -1)
2
(-1, -3)
(-4, -3)
(2, -3)
3
3
2
(-1, -5)
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Graph the following Ellipse
Center (-1, -3)
Major Axis length 6 endpoints (-4, -3)
(2, -3)
Minor Axis length 4 endpoints (-1, -1)
(-1, -5)
Foci
(-1, -1)
Length of LR
2
(-1, -3)
(-4, -3)
(2, -3)
3
3
2
(-1, -5)
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Graph the following Ellipse
9x2 36x 4y2 - 40y - 100 88
(9x2 36x ) (4y2 - 40y ) 88 100
9(x2 4x 22) 4(y2 - 10y (-5)2) 188 36
100
9(x 2)2 4(y - 5)2 324
324
36
81
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Graph the following Ellipse
Center (-2, 5)
a 9 in y direction
b 6 in x direction
(-2, 14)
9
(-2, 5)
(-8, 5)
(4, 5)
6
6
9
(-2, -4)
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Graph the following Ellipse
a 9 b 6
a2 b2 c2
92 62 c2
81 36 c2
(-2, 14)
45 c2
foci
9
(-2, 5)
(4, 5)
(-8, 5)
6
6
9
(-2, -4)
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Graph the following Ellipse
Find theLength of the LR.
a 9 b 6 c
(-2, 14)
9
(-2, 5)
(4, 5)
(-8, 5)
6
6
9
(-2, -4)
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Graph the following Ellipse
Center (-2, 5)
(-2, 14)
Major Axis length 18 endpoints (-2,
14) (-2, -4)
9
Minor Axis length 12 endpoints (-8,
5) (4, 5)
(-2, 5)
(4, 5)
(-8, 5)
6
6
9
Foci
Length of LR
(-2, -4)
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Graph the following Ellipse
4x2 - 32x 25y2 - 150y 189 0
(4x2 8x ) (25y2 - 150y ) 0 - 189
4(x2 - 8x (-4)2) 25(y2 - 6y (-3)2) -189
64 225
4(x - 4)2 25(y - 3)2 100
100
25
4
30
Graph the following Ellipse
Center (4, 3)
a 5 in x direction
b 2 in y direction
(4, 5)
2
(4, 3)
(-1, 3)
(9, 3)
5
5
2
(4, 1)
31
Graph the following Ellipse
a 5 b 2
a2 b2 c2
52 22 c2
25 4 c2
foci
21 c2
(4, 5)
2
(4, 3)
(-1, 3)
(9, 3)
5
5
2
(4, 1)
32
Graph the following Ellipse
a 5 b 2 c
Find the Length of the LR.
(4, 5)
2
(4, 3)
(-1, 3)
(9, 3)
5
5
2
(4, 1)
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Graph the following Ellipse
Center (4, 3)
Major Axis length 10 endpoints (-1, 3)
(9, 3)
Minor Axis length 4 endpoints (4, 5)
(4, 1)
Foci
(4, 5)
Length of LR
2
(4, 3)
(-1, 3)
(9, 3)
5
5
2
(4, 1)
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Graph the following Ellipse
7x2 28x 5y2 - 2y 15 17
(7x2 28x ) (5y2 - 2y ) 17 - 15
7(x2 4x 22) 5(y2 - 2y (-1)2) 2 28 5
7(x 2)2 5(y - 1)2 35
35
5
7
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Graph the following Ellipse
Center (1, -2)
a in y direction
b in x direction
(1, -2)
36
Graph the following Ellipse
a b
a2 b2 c2
foci
(1, -2)
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Graph the following Ellipse
Length of the LR.
(1, -2)
38
Graph the following Ellipse
Center (1, -2)
Major Axis length endpoints
Minor Axis length endpoints
(1, -2)
Foci
Length of LR
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Elliptical Billiard table

• http//cage.ugent.be/hs/billiards/billiards.html
• http//www.cut-the-knot.org/Curriculum/Geometry/En
  velopesInEllipse.shtml

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