### 8.4: Ellipses

```8.4: Ellipses
• Write equations of ellipses
• Graph ellipses
Ellipse
• An ellipse is like an oval.
• Every ellipse has two axes of symmetry
• Called the major axis and the minor axis
• The axes intersect at the center of the ellipse
• The major axis is bigger than the minor axis
• We use c2 = a2 – b2 to find c
• a is always greater b
• The equation is always equal to 1
Ellipses Chart (pg 434)
Standard Form
of Equation
Center
Direction of
Major Axis
Foci
Length of Major
Axis
Length of Minor
Axis
x
- h
a
2
2
+
y
- k
b
2
2
=1
y
- k
a
2
2
+
x
- h
b
2
2
=1
(h,k)
(h,k)
Horizontal
Vertical
(h + c, k), (h – c, k)
(h, k + c), (h, k – c)
2a units
2a units
2b units
2b units
Example One:
Graph the ellipse
x
- 2
4
2
+
y
+ 5
1
2
=1
Graph the ellipse
x
+ 2
81
2
+
y
- 5
16
2
=1
Example Two:
Graph the ellipse
y
- 4
64
2
+
x
- 2
4
2
=1
Graph the ellipse
y
- 2
36
2
+
x
- 4
9
2
=1
Example Three:
Write the equation of the ellipse in the
graph:
Write the equation of the ellipse in the
graph:
Example Four:
Write the equation of the ellipse in the
graph:
Write the equation of
the ellipse in the graph:
Standard Form
Find the coordinates of the center and
foci and the lengths of the major and
minor axes of the ellipse with equation:
x2 + 4y2 + 24y = -32
Standard Form
Find the coordinates of the center and
foci and the lengths of the major and
minor axes of the ellipse with equation:
9x2 + 6y2 – 36x + 12y = 12
```