1. Ellipses
Ellipse: set of all points in a plane such that the sum of the distances from two given points in a plane, called the foci, is constant.
Sum of the distances:
12 units
co-vertex
foci
vertex
vertex
co-vertex
The major axis is the line segment joining the vertices
(through the foci)
The minor axis is the line segment joining the co-vertices (perpendicular to the major axis)

2. Standard Equation of an Ellipse (Center at Origin)
x2 + y2 = 1
a b2
This is the equation
if the major axis is
horizontal.
(0, b)
(–c, 0)
(c, 0)
(–a, 0)
(a, 0)
(0, –b)
The foci of the ellipse lie on the major axis, c units from the
center, where c2 = a2 – b2

3. Standard Equation of an Ellipse (Center at Origin)
x2 + y2 = 1
b a2
This is the equation
if the major axis is
vertical.
(0, a)
(0, c)
(–b, 0)
(b, 0)
(0, –c)
(0, –a)
The foci of the ellipse lie on the major axis, c units from the
center, where c2 = a2 – b2

4. Example: Write an equation of the ellipse whose vertices are (0, –3) and (0, 3) and whose co-vertices are (–2, 0) and (2, 0). Find the foci of the ellipse.
Since the major axis is
vertical, the equation is
the following:
(0, 3)
x2 + y2 = 1 b a2
(0, c)
(–2, 0)
(b, 0)
Since a = 3 b = 2
The equation is x2 + y2 =
(0, –c)
(0, –3)
Use c2 = a2 – b2 to find c.
c2 = 32 – 22
c2 = 9 – 4 = 5
c =
The foci are

5. Example: Write the equation in standard form of 9x2 + 16y2 = 144
Example: Write the equation in standard form of 9x y2 = Find the foci and vertices of the ellipse.
Get the equation in standard form (make it equal to 1):
9x y2 =
Simplify...
x2 + y2 =
That means a = 4 b = 3
Use c2 = a2 – b2 to find c.
c2 = 42 – 32
c2 = 16 – 9 = 7
c =
(0, 3)
(–4,0)
(4, 0)
Vertices:
Foci:
(–c,0)
(c, 0)
(0,-3)