9.4 Ellipse

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9.4 Ellipse
Definition An ellipse is the set of all points
in a plane such that the sum of the distances
from P to two fixed points (F1 and F2) called
foci is constant.
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• A few terms
• Major axis
• 2. Minor axis
• 3. Vertices
• Co-vertices
• 5. Foci
• 6. Center

Longer axis of the ellipse
Shorter axis of the ellipse
Endpoints of the Major axis (as)
Endpoints of the Minor axis (bs)
Always on the Major axis (cs)
Where the Major and Minor intersect (h,k)
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Ellipses can have either a horizontal major axis
or a vertical major axis.
Vertical Major Axis
Horizontal Major Axis
Note When the bigger number is under the x
term, the major axis will be on the x- axis (or
parallel to it if translated). When the bigger
number is under the y term, the major axis will
be on the y-axis (or parallel to it if
translated).
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2a

• The length of the major axis is _______
• The length of the minor axis is ________
• The foci are always on the __________ axis.
• - The following are ___________ true for ellipses

2b
major
ALWAYS
a b c
Vertices Co-vertices Foci
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To write an equation you need the CENTER and the
as and bs.
Example 1 Write the standard form equation for
an ellipse with foci of (0, -4) and (0, 4) and
with minor axis of 6.
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To write an equation you need the CENTER and the
as and bs.
Example 2 Write the standard form equation for
an ellipse with foci of (-8, 0) and (8,0) and
with major axis of 20.
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To write an equation you need the CENTER and the
as and bs.
Example 3 Find the vertices and co-vertices. Is
the ellipse on end or on its side?
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Example 4 Put the following equations in
standard form for an ellipse. State the
vertices, co-vertices, and foci. Does the
ellipse have a horizontal or vertical major axis?
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Example 5 Sketch. Label the center, foci,
vertices and co-vertices.
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Example 6 Write the standard equation for an
ellipse with the given characteristics. a. Foci
(5,0) and (-5, 0 ) b. Co-vertices (0,2) and
(0, -2) Vertices (9, 0) and (-9, 0)
Vertices (3,0) and (-3, 0)
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Example 7 Write a standard form equation for
each ellipse. Identify the center, foci,
vertices, and co-vertices.
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Example 7 Write a standard form equation for
each ellipse. Identify the center, foci,
vertices, and co-vertices.