An ellipse is the collection of points in the plane the sum of whose distances from two fixed points

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An ellipse is the collection of points in the
plane the sum of whose distances from two fixed
points, called the foci, is a constant.
y
Minor Axis
P (x, y)
Major Axis
x
F2
F1
V2
V1
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Theorem Equation of an Ellipse Center at (0,
0) Foci at ( c, 0) Major Axis along the
x-Axis
An equation of the ellipse with center at (0,
0) and foci at (- c, 0) and (c, 0) is
The major axis is the x-axis the vertices are at
(-a, 0) and (a, 0).
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y
F2(c, 0)
F1(-c, 0)
(0, b)
x
V1
(-a, 0)
V2(a, 0)
(0, -b)
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Theorem Equation of an Ellipse Center at (0,
0) Foci at (0, c) Major Axis along the
y-Axis
An equation of the ellipse with center at (0,
0) and foci at (0, - c) and (0, c) is
The major axis is the y-axis the vertices are at
(0, -a) and (0, a).
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y
V2 (0, a)
F2 (0, c)
(b, 0)
(-b, 0)
x
F1 (0, -c)
V1 (0, -a)
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Find an equation of the ellipse with center at
the origin, one focus at (0, 5), and a vertex at
(0, -7). Graph the equation by hand and using a
graphing utility.
Center (0, 0)
Major axis is the y-axis, so equation is of the
form
Distance from center to focus is 5, so c 5
Distance from center to vertex is 7, so a 7
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(0, 7)
FOCI
(0, -7)
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Ellipse with Major Axis Parallel to the x-Axis
where a gt b and b2 a2 - c2.
y
(h c, k)
(h - c, k)
Major axis
(h - a, k)
(h a, k)
(h, k)
x
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Ellipse with Major Axis Parallel to the y-Axis
where a gt b and b2 a2 - c2.
y
(h, k a)
(h, k c)
(h, k)
(h, k - c)
x
Major axis
(h, k - a)
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Center (h, k) (-4, 2)
Major axis parallel to the x-axis
Vertices (h a, k) (-4 3, 2) or (-7, 2) and
(-1, 2)
Foci (h c, k)
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(-4, 4)
V(-1, 2)
V(-7, 2)
F(-6.2, 2)
F(-1.8, 2)
C (-4, 2)
(-4, 0)