1. C O N I C S E C T I O N S
Part 3: The Ellipse

2. Co-vertex
Foci
Ellipse
Vertex
Vertex
Center
Circle
Co-vertex
An ellipse has 2 Focus points that are on the Major Axis and equidistant from the Center of the ellipse.
The Major Axis is the longest segment that cuts the ellipse in half.
It intersects with the ellipse at the Vertices.
The Minor Axis is the shortest segment that cuts the ellipse in half.
It intersects with the ellipse at the Co-vertices.

3. Standard Equation of an Ellipse
(x-h)2 + (y-k)2 = 1
a b2
When major axis is horizontal.
(0, b)
(–c, 0)
(c, 0)
(–a, 0)
(a, 0)
(0, –b)
a = distance from center to vertex
b = distance from center to co-vertex
c = distance from center to focus
c2 = a2 – b2

4. Standard Equation of an Ellipse
(x-h)2 + (y-k)2 = 1
b a2
When major axis is vertical.
(0, a)
(0, c)
(–b, 0)
(b, 0)
(0, –c)
(0, –a)
a = distance from center to vertex
b = distance from center to co-vertex
c = distance from center to focus
c2 = a2 – b2

5. What is the relationship of the denominators?
(x-h)2 + (y-k)2 = 1
a b2
(x-h)2 + (y-k)2 = 1
b a2
(0, a)
(0, b)
(0, c)
(–c, 0)
(c, 0)
(–b, 0)
(b, 0)
(a, 0)
(–a, 0)
(0, –c)
(0, –b)
(0, –a)
a = distance from center to vertex
b = distance from center to co-vertex
c = distance from center to focus
Notice that when the major axis is parallel with the x-axis, a2 goes with the (x-h)2; but when the minor axis is parallel with the x-axis, b2 goes with the (x-h)2
c2 = a2 – b2

6. Mr. Cool Ice Thinks This Stuff is Cool!

7. The equation is (x-0)2 + (y-0)2 = 1 4 9
Write an equation of the ellipse with vertices (0, –3) & (0, 3) and co-vertices (–2, 0) & (2, 0).
(x-h)2 + (y-k)2 = 1
b a2
(0 , 3)
(0, c)
(–2, 0)
(2, 0)
Since a = 3 & b = 2
The equation is (x-0)2 + (y-0)2 = 1
(0, –c)
(0, –3)
Let’s Find the Foci
c2 = a2 – b2 to find c.
c2 = 32 – 22
c2 = 9 – 4 = 5
c =
So the Foci are at:

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

9. Graph (x – 2)2 + (y + 3)2 = 1 25 9 horizontal Center: (2, –3)
a = 5, b = 3
(2, 0)
Start at the center
5 units left and right
(–3,–3)
(7, –3)
3 units up and down
(2,–6)

10. Find center, vertices and foci for the ellipse 36x2 + y2 – 144x + 8y = –124
36x2 – 144x y2 + 8y = –124
Group the x’s and y’s together...
Factor to make the leading coefficients 1
36(x2 – 4x ) + (y2 + 8y ) = –124
+ (36)(4)
+ 16
Complete the squares.
36(x – 2)2 + (y + 4)2 = 36
Set equal to 1
36(x – 2)2 + (y + 4)2 = 36
Center: ( 2, – 4 )
Vertices: ( 2 , 2 ) & (2 , -10 )
(x – 2)2 + (y + 4)2 = 1
Co-vertices: ( 3 , - 4 ) & ( 1, - 4 )
Foci: ( 2 , ) & ( 2, )
Since the major axis is vertical, the vertices will be a units above and below the center.
a = 6 ; b = 1
The foci are c units from the center and c2 = a2 – b2
c2 = 36 – 1
c2 = 35
c =