Ellipse

1
Ellipse

• Conic Sections

2
Ellipse

• The plane can intersect one nappe of the cone at
  an angle to the axis resulting in an ellipse.

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Ellipse - Definition
An ellipse is the set of all points in a plane
such that the sum of the distances from two
points (foci) is a constant.
d1 d2 a constant value.
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Finding An Equation

• Ellipse

5
Ellipse - Equation
To find the equation of an ellipse, let the
center be at (0, 0). The vertices on the axes
are at (a, 0), (-a, 0),(0, b) and (0, -b). The
foci are at (c, 0) and (-c, 0).
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Ellipse - Equation
According to the definition. The sum of the
distances from the foci to any point on the
ellipse is a constant.
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Ellipse - Equation
The distance from the foci to the point (a, 0) is
2a. Why?
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Ellipse - Equation
The distance from (c, 0) to (a, 0) is the same as
from (-a, 0) to (-c, 0).
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Ellipse - Equation
The distance from (-c, 0) to (a, 0) added to the
distance from (-a, 0) to (-c, 0) is the same as
going from (-a, 0) to (a, 0) which is a distance
of 2a.
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Ellipse - Equation
Therefore, d1 d2 2a. Using the distance
formula,
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Ellipse - Equation
Simplify
Square both sides.
Subtract y2 and square binomials.
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Ellipse - Equation
Simplify
Solve for the term with the square root.
Square both sides.
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Ellipse - Equation
Simplify
Get x terms, y terms, and other terms together.
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Ellipse - Equation
Simplify
Divide both sides by a2(c2-a2)
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Ellipse - Equation
Change the sign and run the negative through the
denominator.
At this point, lets pause and investigate a2
c2.
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Ellipse - Equation
d1 d2 must equal 2a. However, the triangle
created is an isosceles triangle and d1 d2.
Therefore, d1 and d2 for the point (0, b) must
both equal a.
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Ellipse - Equation
This creates a right triangle with hypotenuse of
length a and legs of length b and c. Using
the pythagorean theorem, b2 c2 a2.
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Ellipse - Equation
We now know..
and b2 c2 a2
b2 a2 c2
Substituting for a2 - c2
where c2 a2 b2
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Ellipse - Equation
The equation of an ellipse centered at (0, 0) is
.
where c2 a2 b2 andc is the distance from
the center to the foci.
Shifting the graph over h units and up k units,
the center is at (h, k) and the equation is
where c2 a2 b2 andc is the distance from
the center to the foci.
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Ellipse - Graphing
where c2 a2 b2 andc is the distance from
the center to the foci.
Vertices are a units in the x direction and b
units in the y direction.
b
a
a
c
c
The foci are c units in the direction of the
longer (major) axis.
b
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Graph - Example 1

• Ellipse

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Ellipse - Graphing
Graph
Center
(2, -3)
Distance to vertices in x direction
4
Distance to vertices in y direction
5
Distance to foci
c216 - 25 c2 9 c 3
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Ellipse - Graphing
Graph
Center
(2, -3)
Distance to vertices in x direction
4
Distance to vertices in y direction
5
Distance to foci
c216 - 25 c2 9 c 3
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Graph - Example 2

• Ellipse

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Ellipse - Graphing
Graph
Complete the squares.
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Ellipse - Graphing
Graph
Center
(-1, 3)
Distance to vertices in x direction
Distance to vertices in y direction
5
Distance to foci
c225 - 10 c2 15 c
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Find An Equation

• Ellipse

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Ellipse Find An Equation
Find an equation of an ellipse with foci at (-1,
-3) and (5, -3). The minor axis has a length of
4.
The center is the midpoint of the foci or (2,
-3).
The minor axis has a length of 4 and the vertices
must be 2 units from the center.
Start writing the equation.
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Ellipse Find An Equation
c2 a2 b2. Since the major axis is in the x
direction, a2 gt 4
9 a2 4
a2 13Replace a2 in the equation.
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Ellipse Find An Equation
The equation is
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Ellipse Table
Center
(h, k)
Vertices
Foci
c2 a2 b2
If a2 gt b2
If b2 gt a2