Transcript 6 -6 Factoring by Grouping

4 minutes
Warm-Up
Write the standard equation of the circle with
the given radius and center.
1) 9; (0,0)
2) 1; (0,5)
3) 4; (-8,-1)
4) 5; (4,2)
9.4.1 Ellipses
Objectives:
•Write the standard equation for an ellipse given
sufficient information
•Given an equation of an ellipse, graph it and label the
center, vertices, co-vertices, and foci
Definition of Ellipse
An ellipse is the set of all points P in a plane
such that the sum of the distances from P to
two fixed points, F1 and F2, called the foci, is
a constant.
P
F1
F
2
F1P + F2P = 2a
Standard Equation of an Ellipse
Horizontal Major Axis:
x2
+
2
a
y2
b2
y
=1
V1(–a, 0) (0, b)
a2 > b2
a2 – b2 = c2
O
F1(–c, 0) (0, –b)
length of major axis: 2a
length of minor axis: 2b
V2 (a, 0)
x
F2 (c, 0)
Standard Equation of an Ellipse
Vertical Major Axis:
x2
+
2
b
y2
a2
=1
–
b2
V2 (0, a)
F2 (0, c)
a2 > b2
a2
y
=
c2
(–b, 0)
O
length of major axis: 2a
length of minor axis: 2b
F1(0, –c)
(b, 0)
x
V1(0, –a)
Example 1
Write the standard equation for an ellipse with
foci at (-8,0) and (8,0) and with a major axis of
20. Sketch the graph.
length of major axis: 2a
2a = 20, so a = 10
8
6
4
2
a2 – b2 = c2
102 – b2 = 82
b2 = 100 - 64
-8 -6 -4 -2-2
-4
-6
-8
b2 = 36, so b = 6
x2
y2
+
=1
100
36
2 4 6 8
Example 2
Find the vertices and co-vertices of the ellipse.
x2
y2
+
=1
16
49
vertices: (0,7) and (0,-7)
co-vertices: (4,0) and (-4,0)
Example 3
Write the standard equation of the ellipse.
length of major axis: 2a
2a = 16, so a = 8
length of minor axis: 2b
2b = 8, so b = 4
x2
y2
+
=1
16
64
8
6
4
2
-8 -6 -4 -2-2
-4
-6
-8
2 4 6 8
Practice
Write the standard equation for an ellipse
with foci at (5,0) and (-5,0) and with vertices
at (9,0) and (-9,0). Sketch the graph.
Homework
p.591 #11,13,21,27,29,41,43,45
Quiz Tomorrow (9.1-9.3)
5 minutes
Warm-Up
Write the standard equation for an ellipse with
foci at (-5,0) and (5,0) and with a major axis of
18. Sketch the graph.
9.4.2 Ellipses
Objectives:
•Write the standard equation for an ellipse given
sufficient information
•Given an equation of an ellipse, graph it and label the
center, vertices, co-vertices, and foci
Standard Equation of a Translated Ellipse
Horizontal Major Axis:
(x – h)2
(y – k)2
+
=1
2
2
a
b
a2 > b2
a2 – b2 = c2
length of major axis: 2a
length of minor axis: 2b
Standard Equation of a Translated Ellipse
Vertical Major Axis:
(x – h)2
(y – k)2
+
=1
2
2
b
a
a2 > b2
a2 – b2 = c2
length of major axis: 2a
length of minor axis: 2b
Example 1
An ellipse is defined by the equation 4x2 + 9y2 – 16x +
18y = 11. Write the standard equation and identify the
coordinates of the center, vertices, co-vertices, and
foci. Sketch the graph of the ellipse.
4x2 – 16x + 9y2 + 18y = 11
4(x2 – 4x) + 9(y2 + 2y) = 11
4(x2 – 4x + 4) + 9(y2 + 2y + 1) = 11 + 4(4) + 9(1)
4(x – 2)2 + 9(y + 1)2 = 36
4(x  2)2 9(y  1)2 36


36
36
36
(x  2)2 (y  1)2

1
9
4
Example 1
An ellipse is defined by the equation 4x2 + 9y2 – 16x +
18y = 11. Write the standard equation and identify the
coordinates of the center, vertices, co-vertices, and
foci. Sketch the graph of the ellipse.
(x  2)2 (y  1)2

1
9
4
6
center: (2,-1)
a2 = 9, so a = 3
vertices: (-1,-1) and (5,-1)
b2 = 4, so b = 2
co-vertices: (2,1) and (2,-3)
a2 – b2 = c2
9 - 4 = c2
c 5
4
2
-6
-4
-2
-2
2
4
6
-4
-6
foci : (2  5, 1) & (2  5, 1)
Practice
Write the standard equation for the ellipse
9x2 + 16y2 – 36x – 64y – 44 = 0. Identify the
center, vertices, co-vertices, and foci.
Homework
p.591 #19,23,25,31,33,37,51,55