11/26/2012
Section 9.1 The Ellipse
1
11/26/2012
The relationship of a, b and c in an Ellipse Used to Find the Foci, c.
(0,b)
2
a
1
b
−4
−3
−2
c
−1
1
2
(c,0)
3
4
−1
−2
3
Applying the Pythagorean Theorem we get
b 2 + c 2 = a 2 or c 2 = a 2 − b 2
2
11/26/2012
X and Y Intercepts of an Ellipse
Ellipse with the Major Axis on the X‐Axis
3
11/26/2012
Ellipse with the Major Axis on the Y‐Axis
Graph and locate the foci.
9x 2 + 4 y 2 = 36
4
y
3
2
1
x
−4
−3
−2
−1
1
2
3
4
5
−1
−2
−3
−4
4
11/26/2012
Graph and locate the foci.
x2 y2
+
=1
81 25
4 16
4
y
3
2
1
x
−4
−3
−2
−1
1
2
3
4
5
−1
−2
−3
−4
Graph and give the location of the foci.
( x -1) 2 ( y + 2) 2
+
=1
16
9
4
y
3
2
1
x
−4
−3
−2
−1
1
2
3
4
5
−11
−2
−3
−4
5
11/26/2012
Write the standard equation
of the ellipse pictured.
y
4
3
2
1
−3
−2
−1
1
2
3
−1
−22
−3
−4
Standard form of Equations of Ellipses Centered at h,k
6
11/26/2012
How to use the Center to Find the Endpoints of the Axes
7
11/26/2012
( x + 2) 2 ( y − 1) 2
+
=1
9
4
Where are the foci?
Where are the vertices, and
Graph
endpoints of the minor axis.
axis
4
y
3
2
1
x
−4
−3
−2
−1
1
2
3
4
5
−11
−2
−3
−4
( x − 3) 2 ( y − 1) 2
+
=1
8
9
Where are the foci?
Where are the vertices, and
Graph
endpoints of the minor axis.
axis
4
y
3
2
1
x
−4
−3
−2
−1
1
2
3
4
5
−11
−2
−3
−4
8
11/26/2012
Write the equation in standard form of an ellipse given the following information.
Find the center of the ellipse, and the endpoints of the minor axis.
Vertices: (-6,0), (6, 0) Foci: (-2, 0),(2, 0)
Write the equation in standard form of an ellipse given the following information.
Vertices: (0, -4), (0, 4) Foci: (0, -3),(0, 3)
9
11/26/2012
Write the equation in standard form of an ellipse given the following information
Major axis horizontal with lenth 12; length of minor axis 6; center (0, 0)
Write the standard form of the equation of the ellipse satisfying the given conditions.
Major axis is vertical with length 20; length of the minor axis is 10; center (2, ‐3)
10
11/26/2012
Write the standard form of the equation of the ellipse satisfying the given conditions.
Endpoints of the major axis are (2, 2) and (8, 2); endpoints of the minor axis are (5,3) and (5, 1).
Completing The Square to Find the Standard Form of the Ellipse
11
11/26/2012
Convert each equation to standard form by completing the square on x and y. Then graph and give location of foci.
4x2 + 9y2 – 32x + 36y + 64
12