Conics Instructional Unit

 

Day 5  - Ellipses

 

by

 

Mandy Stein

 

Ellipse

 

The locus of all points P(x,y) such that the sum of the distance from P to two fixed points, called foci, are constant.

 

 

 

                                                         

 

 

Major axis – the longer axis of the ellipse

Vertices – endpoints of the minor axis

Minor axis – the shorter axis of the ellipse

Co-Vertices – the endpoints of the minor axis

 

Standard Equation of an Ellipse

Standard equation of an ellipse centered at (h , k)

 

Horizontal Major Axis

 +  = 1

 

Vertices: (h – a, k) & (h + a, k)

Co-Vertices: (h, k + b) & (h, k – b)

Foci: (h – c, k) & (h + c, k)

Major axis = 2a

Minor axis = 2b

a2 > b2

a2 - b2 = c2

Vertical Major Axis

 +  = 1

 

Vertices: (h, k – a) & (h, k + a)

Co-Vertices: (h – b, k) & (h + b, k)

Foci: (h, k – c) & (h, k + c)

Major axis = 2a

Minor axis = 2b

a2 > b2

a2 - b2 = c2

 

           Horizontal Major Axis

 

 

 

              Vertical Major Axis

 

 

 

Eccentricity – a measure of how round or flat an ellipse is.  The eccentricity, E, is a ratio between the distance, c, between the center and a focus to the distance, a, between the center and a vertex.  As e approaches 1, the ellipse becomes flatter.

E =

 

To graph an ellipse:

  1. Determine if the ellipse has a horizontal or vertical major axis
  2. Identify the vertices
  3. Identify the co-vertices
  4. Identify the foci
  5. Graph the vertices, co-vertices, foci, and sketch the ellipse

 

 

 +  = 1

Vertical major axis

Vertices: (-2,9) & (-2,11)

Co-vertices: (-10,-1) & (6,-1)

Foci: (-2,5) & (-2,-7)

 

 

 

 +  = 1

Horizontal major axis

Vertices: (8,2) & (-2,2)

Co-vertices: (3,5) & (3,1)

Foci: (7,2) & (-1,2)

 

 

To graph an equation not in standard form:

  1. Determine if the ellipse has a vertical or horizontal major axis
  2. Write the equation in standard form by completing the square
  3. Identify the vertices
  4. Identify the co-vertices
  5. Identify the foci
  6. Graph the vertices, co-vertices, foci, and sketch the ellipse

 

 

4x2 + y2 +24x – 4y + 36 = 0

 

First, we put the equation in standard form by completing the square

4x2 + y2 +24x – 4y = -36

4(x2 + 6x) + (y2 – 4y) = -36

4(x2 + 6x + 9) + (y2 – 4y + 4) = -36 + 4(9) + 4

4(x + 3) 2 + (y – 2) 2 = 4

 +  = 1

Vertical major axis

Center: (-3,2)

Vertices: (-3,0) & (-3,4)

Co-vertices: (-4,2) & (-2,2)

Foci: (-3,2-) & (-3,2 + )

 

 

 

To write the equation of an ellipse:

  1. Determine if the ellipse has a horizontal or vertical major axis
  2. Identify the center
  3. Identify the vertices
  4. Identify the co-vertices
  5. Obtain the value of h, k, a, b, and c
  6. Substitute h, k, a, and b into the correct equation and simplify

 

Horizontal major axis

Center: (2,-4)

Vertices: (-3,-4) & (7,-4)

Co-vertices: (2,-1) & (2,-7)

2a = 10; a = 5

2b = 6; b = 3

 +  = 1

 

 

Vertical major axis

Center: (3,0)

Vertices: (0,0) & (6,0)

Co-vertices: (3,6) & (3,-6)

2a = 6; a = 3

2b = 12; b = 6

 +  = 1

 

Day 6 - Circles

 

Return to my EMAT 6690 page