C O N I CS E C T I O N S

Part 3: The Ellipse

Circle

EllipseFoci

Vertex

Co-vertex

The Major Axis is the longest segment that cuts the ellipse in half.

The Minor Axis is the shortest segment that cuts the ellipse in half.

Vertex

It intersects with the ellipse at the Vertices.

It intersects with the ellipse at the Co-vertices.

Co-vertex

An ellipse has 2 Focus points that are on the Major Axis and equidistant from the Center of the ellipse.

Center

Standard Equation of an Ellipse

(x-h)2 + (y-k)2 = 1

a2 b2

When major axis is horizontal.

a = distance from center to vertexb = distance from center to co-vertexc = distance from center to focus

c2 = a2 – b2

(c, 0) (–c, 0) (–a, 0) (a, 0)

(0, b)

(0, –b)

Standard Equation of an Ellipse

When major axis is vertical. (x-h)2 + (y-k)2 = 1

b2 a2

a = distance from center to vertexb = distance from center to co-vertexc = distance from center to focus

c2 = a2 – b2

(0, c)

(0, –c)

(0, –a)

(0, a)

(b, 0) (–b, 0)

What is the relationship of the denominators?

(x-h)2 + (y-k)2 = 1

a2 b2

a = distance from center to vertexb = distance from center to co-vertexc = distance from center to focus

(c, 0) (–c, 0)

(–a, 0) (a, 0)

(0, b)

(0, –b)

c2 = a2 – b2

(0, c)

(0, –c)

(0, –a)

(0, a)

(b, 0) (–b, 0)

(x-h)2 + (y-k)2 = 1

b2 a2

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

Mr. Cool Ice Thinks This Stuff is Cool!

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

b2 a2

c2 = a2 – b2 to find c. c2 = 32 – 22

c2 = 9 – 4 = 5c =

(0, c)

(0, –c)

(0, –3)

(0 , 3)

(2, 0) (–2, 0)

5

Since a = 3 & b = 2The equation is (x-0)2 + (y-0)2 = 1 4 9

0, 5 and 0, 5

Let’s Find the Foci

So the Foci are at:

Example: Write 9x2 + 16y2 = 144 in standard form. Find the foci and vertices.

9x2 + 16y2 = 144144 144 144

Use c2 = a2 – b2 to find c. c2 = 42 – 32

c2 = 16 – 9 = 7c =

(c, 0) (–c,0)

(–4,0) (4, 0)

(0, 3)

(0,-3)

7

That means a = 4 b = 3 Vertices:Foci:

4,0 and 4,0

Simplify...

x2 + y2 = 116

9

0,7 0,7 and

horizontalCenter: (2, –3)a = 5, b = 3

Graph (x – 2)2 + (y + 3)2 = 1 25 9

(2, 0)

(2,–6)

(–3,–3) (7, –3)

Start at the center

5 units left and right

3 units up and down

Find center, vertices and foci for the ellipse 36x2 + y2 – 144x + 8y = –124

36(x – 2)2 + (y + 4)2 = 36

Group the x’s and y’s together...36x2 – 144x + y2 + 8y = –124

Factor to make the leading coefficients 1

36(x2 – 4x ) + (y2 + 8y ) = –124 Complete the squares. + 4 +16 + (36)(4)

Set equal to 1

36(x – 2)2 + (y + 4)2 = 36 36 36 36

(x – 2)2 + (y + 4)2 = 1 1 36

Center: ( 2, – 4 )

Since the major axis is vertical, the vertices will be a units above and below the center.

Vertices: ( 2 , 2 ) & (2 , -10 )

+ 16

The foci are c units from the center and c2 = a2 – b2

c2 = 36 – 1c2 = 35c =

a = 6 ; b = 1

Co-vertices: ( 3 , - 4 ) & ( 1, - 4 )

Foci: ( 2 , - 4 - ) & ( 2, - 4 + )

35

35 35