9.4 Graph & Write Equations
of Ellipses
Algebra II
• An ellipse is a set of points
such that the distance between
that point and two fixed points
called Foci remains constant
d1
d2
f1
f2
d4
d3
d1 + d2 = d3 + d4
cv1
v1
F1
c
cv2
F2
v2
• The line that goes through the Foci is the Major
Axis.
• The midpoint of that segment between the foci is
the Center of the ellipse (c)
• The intersection of the major axis and the ellipse
itself results in two points, the Vertices (v)
• The line that passes through the center and is
perpendicular to the major axis is called the
Minor Axis
• The intersection of the minor axis and the ellipse
results in two points known as co-vertices
Example of ellipse with vertical
major axis
Example of ellipse with horizontal
major axis
Standard Form for Elliptical
Equations
Major
Axis
Equation
(length is
2a)
Minor Axis
(length is
2b)
x2 y2
 2 1
2
a
b
Horizontal Vertical
Vertices CoVertices
(a,0) (-a,0)
(0,b) (0,-b)
(0,a) (0,-a)
(b,0) (-b,0)
Note that a is the biggest number!!!
x2 y2
 2  1 Vertical
2
b
a
Horizontal
• The foci lie on the major axis at the
points:
• (c,0) (-c,0) for horizontal major axis
• (0,c) (0,-c) for vertical major axis
2
• Where c
=
2
a
–
2
b
1.) Write the equation of an ellipse
with center (0,0) that has a vertex
at (0,7) & co-vertex at (-3,0)
• Since the vertex is on the y axis (0,7) a=7
• The co-vertex is on the x-axis (-3,0) b=3
• The ellipse has a vertical major axis & is of the
form
2
2
x
y
 2 1
2
b
a
2
2
x
y

1
9 49
Ex. 2) Given the equation
9x2 + 16y2 = 144
Identify:foci, vertices, & co-vertices
• First put the equation in standard form:
2
2
9 x 16 y
144


144 144 144
2
2
x
y

1
16 9
2
2
x
y

1
16 9
• From this we know the major axis is
horizontal & a=4, b=3
• So the vertices are (4,0) & (-4,0)
•
the co-vertices are (0,3) & (0,-3)
• To find the foci we use c2 = a2 – b2
•
c2 = 16 – 9
•
c = √7
• So the foci are at (√7,0) (-√7,0)
Graph Ex. 2
•
Ex. 3 Write an equation of the ellipse given a
vertex & a focus & a center at (0,0)
• Vertex: (7,0)
• Co-vertex: (0,2)
Standard Form for Elliptical
Equations
Major
Axis
Equation
(length is
2a)
Minor Axis
(length is
2b)
x2 y2
 2 1
2
a
b
Horizontal Vertical
Vertices CoVertices
(a,0) (-a,0)
(0,b) (0,-b)
(0,a) (0,-a)
(b,0) (-b,0)
Note that a is the biggest number!!!
x2 y2
 2  1 Vertical
2
b
a
Horizontal
Ex. 4 Write an equation of the ellipse given a
vertex & a focus & a center at (0,0)
• Vertex: (0,10)
• Focus: (0,8)
Ex. 5 Write an equation of the ellipse given a
vertex & a focus & a center at (0,0)
• Co-vertex: 0,5
• Focus: (-15,0)
7

Assignment