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

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 .

Standard Equation of an Ellipse

(x-h)

2

a

2 + (y-k) 2

b

2 = 1

When major axis is horizontal.

(0, b) (–a, 0) (–c, 0) (c, 0) (a, 0) (0, –b)

a b = distance from center to vertex = distance from center to co-vertex c = distance from center to focus c

2 =

a

2 –

b

2

Standard Equation of an Ellipse

(x-h)

2

b

2 + (y-k) 2

a

2 = 1

When major axis is vertical.

(0, a) (0, c) (–b, 0) (b, 0) (0, –c) (0, –a)

a b = distance from center to vertex = distance from center to co-vertex c = distance from center to focus c

2 =

a

2 –

b

2

a

2

What is the relationship of the denominators?

(x-h)

2 + (y-k)

b

2 2 = 1

(x-h)

2

b

2 + (y-k)

a

2 2 = 1 (0, a) (0, b) (–c, 0) (c, 0) (–b, 0) (0, c) (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

c 2 = a 2 – b 2

Notice that when the major axis is parallel with the x-axis, a 2 goes with the (x-h) 2 ; but when the minor axis is parallel with the x-axis, b 2 goes with the (x-h) 2

Mr. Cool Ice Thinks This Stuff is Cool!

Write an equation of the ellipse with vertices (0, –3) and co-vertices ( –2, 0) & (2, 0) .

& (0, 3)

(–2, 0) (0 , 3) (0, c) (2, 0) (0, –3) (0, –c)

Let’s Find the Foci

c

2

c

2

c

2 = = = 9 – 4 = 5

c

= 

a

3 2 2 5 – –

b

2 2 2

to find c.

(

x-h)

2

b

2 + (

y-k) a

2 2 = 1 Since a = 3 4 9 & b = 2 The equation is (x-0) 2 + (y-0) 2 = 1 So the Foci are at:  

and

  5 

Example:

Write

9x 2 + 16y 2 = 144

in standard form

.

Find the foci and vertices.

9

x

2 144

x

2 + 16

y

2 144 + y 2 = 144 = 1 144 16 9

Simplify...

Use c

2

c

2

c

2 = =

a

4 2 2 – –

b

3 2 2

to find c

= 16 – 9 = 7 =

 7

c .

That means

a = 4 b = 3

Foci:

   4,0 

and

7 , 0 

and

 

(0, 3) (–4,0) (–c,0) (c, 0) (4, 0) (0,-3)

Graph (

x

– 2) 2 + (

y

+ 3) 25 9 2 = 1

horizontal

Center: (2, –3) a = 5

,

b = 3

Start at the center

5

units left and right

3

units up and down

(–3,–3) (2, 0) (2,–6) (7, –3)

Find center, vertices and foci for the ellipse 36

x

2 +

y

2 – 144

x

+ 8

y

= –124 36

x

2 – 144

x

+

y

2 + 8

y

= –124

Group the x’s and y’s together...

36(

x

2 – 4

x Factor to make the leading coefficients 1

) + (

y

2 + 8

y

+16 ) = –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 36 36 36 Center: ( 2, – 4 ) Vertices: ( 2 , 2 ) & (2 , -10 ) (

x

– 2) 2 1 + (

y

+ 4) 36 2 = 1 Co-vertices: ( 3 , - 4 ) & ( 1, - 4 ) Foci: ( 2 , - 4 -

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 c

2

= units from the center and c

36

–

1 2

c

2

=

35 c =  35

= a

2

– b

2 35