Math 102
8.2 "Ellipses"
Objectives:
* Find the vertices, endpoints of the minor axis, and the foci of an ellipse.
* Sketch the graph of an ellipse.
* Write the equation of an ellipse in standard form.
* Determine the equation of an ellipse.
Ellipses Centered at (0,0)
De…nition:
Ellipse
An ellipse is the set of points in a plane such that the sum of their distances from two …xed points is a constant.
Each …xed point is called a focus (plural: foci) of the ellipse.
Equation of an Ellipse with Center (0,0) and Horizontal Major Axis
y2
x2
The graph of the equation
+ 2 = 1 for a2 > b2 is an ellipse centered at the origin.
2
a
b
The endpoints of its major axis (vertices or x intercepts) are at (a; 0) and ( a; 0) :
The endpoints of its minor axis (y
intercepts) are at (0; b) and (0; b)
The foci are at (c; 0) and ( c; 0) ; where c2 = a2
b2 :
Equation of an Ellipse with Center (0,0) and Vertical Major Axis
x2
y2
The graph of the equation
+ 2 = 1 for a2 > b2 is an ellipse centered at the origin.
2
b
a
The endpoints of its major axis (vertices or y intercepts) are at (0; a) and (0; a) :
The endpoints of its minor axis (x
intercepts) are at (b; 0) and ( b; 0)
The foci are at (0; c) and (0; c) ; where c2 = a2
b2 :
Example 1: (Graphing ellipses)
Find the vertices, the endpoints of the minor axis, and the foci of each ellipse, and sketch its graph.
x2
y2
a)
+
=1
25
9
y
6
4
2
-6
-4
-2
2
4
-2
-4
-6
Page: 1
Notes by Bibiana Lopez
6
x
College Algebra by Kaufmann and Schwitters
b)
8.2
y2
x2
+
=1
4
16
y
4
2
-4
-2
2
4
-2
-4
Example 2: (Finding equations of ellipses)
Find an equation of the ellipse that satis…es the given conditions.
a) Vertices ( 4; 0) ; Foci ( 2; 0)
b) Vertices (0; 3) ; Foci (0; 2)
Other Ellipses
Equation of an Ellipse with Center (h; k) and Horizontal Major Axis
The graph of the equation
for a2 > b2 is an ellipse centered at (h; k) .
The endpoints of its major axis are at (h + a; k) and (h
a; k) :
The endpoints of its minor axis are at (h; k + b) and (h; k
The foci are at (h + c; k) and (h
2
2
c; k) ; where c = a
b)
2
b :
Page: 2
Notes by Bibiana Lopez
x
College Algebra by Kaufmann and Schwitters
8.2
Equation of an Ellipse with Center (h; k) and Vertical Major Axis
The graph of the equation
for a2 > b2 is an ellipse centered at (h; k) .
The endpoints of its major axis are at (h; k + a) and (h; k
The endpoints of its minor axis are at (h + b; k) and (h
The foci are at (h; k + c) and (h; k
c) ; where c2 = a2
a) :
b; k)
b2 :
Example 3: (Graphing ellipses)
Find the vertices, the endpoints of the minor axis, and the foci of each ellipse, and sketch its graph.
a) 4x2
8x + 9y 2
36y + 4 = 0
y
6
4
2
-6
-4
-2
2
4
6
2
4
6
x
-2
-4
-6
b) 9x2
36x + 4y 2 + 16y + 16 = 0
y
6
4
2
-6
-4
-2
-2
-4
-6
Page: 3
Notes by Bibiana Lopez
x