4th Nine Weeks
Week 4.6
Ellipses
2A.5.B
I. Warm up
If c2 = a2 – b2, find c if
1. a = 13, b = 5
2. a = 4, b = 3
Algebra 2
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4th Nine Weeks
Week 4.6
I. Lesson notes
An ellipse
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___________________________________________________________________________
The foci
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This distance d can be represented by the length of a piece of string connecting two pushpins
located at the foci.
Example 1: Using the Distance Formula to Find the Constant Sum of an Ellipse
Find the constant sum for an ellipse with foci F1 (-3, 0) and F2 (3, 0) and the point on the ellipse
(0, 4) .
Try it: Find the constant sum for an ellipse with foci F 1 (0, -8)and F 2 (0, 8) and the point on
the ellipse (0, 10).
Algebra 2
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4th Nine Weeks
Week 4.6
major axis __________________________________________________________________
vertices of the ellipse _________________________________________________________
minor axis __________________________________________________________________
co-vertices of the ellipse _______________________________________________________
The major axis and minor axis are perpendicular and intersect at the center of the ellipse.
The standard form of an ellipse centered at (0, 0) depends on whether the major axis is
horizontal or vertical.
The values a, b, and c are related by the equation c2 = a2 – b2 . Also note that the length of the
major axis is 2a, the length of the minor axis is 2b, and a > b.
Algebra 2
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4th Nine Weeks
Week 4.6
Example 2: Using Standard Form to Write an Equation for an Ellipse
Write an equation in standard form for each ellipse with center (0, 0).
A

Choose the appropriate form of equation.

Identify the values of a and c.

Use the relationship c2 = a2 – b2 to find b2.

Write the equation.
B the ellipse with vertex (0, 8) and co-vertex (3, 0)

Choose the appropriate form of equation.

Identify the values of a and b.

Write the equation.
Try it: Write an equation in standard form for each ellipse with center (0, 0) .
Vertex (9, 0) and co-vertex (0, 5)
Algebra 2
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Week 4.6
Ellipses may also be translated so that the center is not the origin.
Example 3: Graphing Ellipses
( x  3) 2 ( y  1) 2

1
Graph the ellipse
16
36

Rewrite the equation as

Identify the values of h, k, a, and b.

Get the vertices and co-vertices
Try it: Graph the ellipse
Algebra 2
( x  2) 2 ( y  4) 2

1
25
9
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Week 4.6
NAME_____________________________________DATE_________________PER._______
Independent Practice: Ellipses
Find the constant sum of an ellipse with the given foci and point on the ellipse.
1. F1(40, 0), F2(-40, 0), P(0, -9)
2. F1(0,-20), F2(0,20), P(15,0)
Write an equation in standard form for each ellipse with center (0, 0) .
3 Vertex (15, 0), focus (9, 0)
4. Co-vertex (0, -21), focus (-75, 0)
5. Co-vertex (-20, 0), focus (0, 48)
6. Vertex (61, 0), focus (60, 0)
Algebra 2
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4th Nine Weeks
7. Co-vertex (4, 0) , focus (0, -3)
Week 4.6
8. Vertex (5, 0), co-vertex (0, -2)
Graph each ellipse
( x  3)2 ( y  2) 2

1
9.
9
16
( x  4) 2 ( y  1) 2

1
10.
36
25
( x  2) 2 ( y  4) 2

1
11.
25
16
( y  2) 2 ( x  2) 2

1
12.
4
9
Algebra 2
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