What is the set of all points in a plane in which the sum of the distances from two fixed points is a constant?

Conic Sections

Ellipses: 

·        An ellipse is a set of points in a plane such that sum of the distances from each point to two set points called the foci is constant.  If you fixed two points in a plane and tied a string to each of these points leaving slack in the string and pulled it taut tracing in a loop, you would form an ellipse. The two fixed points to which the string was fixed would be the foci.

·        The Major Axis is the longer axis through the ellipse. The Minor Axis is the shorter axis through the ellipse. The vertices are the endpoints on these axes.

·        The eccentricity of an ellipse is between 0 and 1, but not ever equal to 0 or 1. Eccentricity is the value c/a.  The book uses the letter e for eccentricity.  Do not confuse this with e » 2.7181…. The more circular the ellipse is, the closer the eccentricity is to 0. The more elongated the ellipse is, the closer the eccentricity is to 1. 

·        c is the distance between the center point and a focus. a is the distance between the center point and an end point along the major axis.  b is the distance between the center point and an endpoint along the minor axis.

·        By the way, c2 = a2 – b2    In the examples: c = 7, a = 8 and b = Ö15 for the wide example . Note: 49 = 64 – 15.  In the other example, c = 5, a = 7, and

      b = Ö24; then 25 = 49 – 24.

·        If a2 is below the x term in the formula, then the ellipse is wider in the horizontal direction than it is long in the vertical direction.

     In the example below for the ellipse which is wider than it is tall:      

·        If a2 is below the y term in the formula, then the ellipse is longer in the vertical direction than it is in the horizontal direction.

            In the example below for the ellipse that is taller than it is wide:           

·        The forms of the equations that were given is the standard form. In this form we can easily pick out the center, lengths of the major and minor axes, then the foci, vertices, etc.

Let’s Practice:

State the center, the length and direction of the major axis, the length and direction of the minor axis, the foci, all four vertices, and the eccentricity. 

1.

2.

3.

4.

5.

·        Like all conic sections, ellipses can be expressed in the general form

      Ax2 + By2 + Cx + Dy + Exy + F = 0.

            Multiply out the squares, collect like terms and set the equation to 0 to put the

            following ellipse in this form.

Practice rewriting into General Form.

6.

7.

·        Now work backwards. We will also complete the square to find the equation of an ellipse which illustrates the location of the center and the shape of the ellipse. 

Example:  x2 + 9y2 –6x + 90y = -225 Û (x2 – 6x      ) + 9(y2  + 10y    ) = -225 Û

(x2 – 6x + 9) + 9(y2 + 10y + 25) = -225 +9 + 225 Û  ( x – 3)2 + 9(y + 5)2 = 9 Û  

This is an ellipse with center (3, -5) .  The length of the major axis is 6 and is horizontal.  The length of the minor axis is 2 and is vertical.  The major vertices are at (0, -5) and (6, -5).  c = Ö(9 – 1) = Ö8.  The foci are at ( 3 – Ö8, -5) and ( 3 + Ö8 , -5).

This ellipse has eccentricity 0.9428. It is relatively narrow.

More Practice:  Complete the square to rewrite the equation of the ellipse in standard form.  State the center and the lengths of the major and minor axes.

8.  x2 + 2y2 - 10x + 8y + 29 = 0

9.  4x2 + 5y2 + 16x – 20y + 31 = 0

10. 2x2 + 3y2 – 6 = 0 

This is an ellipse with center (3, -5) .  The length of the major axis is 6 and is horizontal.  The length of the minor axis is 2 and is vertical.  The vertices are at (0, -5) and (6, -5).  c = Ö(9 – 1) = Ö8.  The foci are at

( 3 – Ö8, -5) and ( 3 + Ö8 , -5).   This ellipse has eccentricity 0.9428. It is relatively narrow.

·        To Graph the ellipse in this example:

Practice graphing ellipses:

11.  Rewrite 2x2 + 3y2 – 6 = 0  as two functions in order to graph the ellipse on the graphing calculator.

12. Rewrite

13. Rewrite 

·        We will only deal with ellipses with horizontal or vertical axes. Ellipses can be tilted in the plane. If an ellipse has such a rotation, it will have an Bxy term in its general form.  If you are interested, you can investigate the topic of rotations of conics on the Internet.

·        Applications of Ellipses:  Orbits, wings of some planes, echo halls....

Wrap up Problems:

14. Look at the ellipses a) through d) in 19) through 22) page 540 of your text.  List the letters of the ellipses in order of eccentricity from greatest to least.

15. #31 page 541. Major vertices (-7, 0), (7, 0)  and foci (-3, 0), (3, 0).  Find an equation of an ellipse in standard form which satisfies this criteria.

16. #35 page 541. Major vertices (-2, 0), (2, 0)  and length of major axis is 6.  Find an equation of an ellipse in standard form which satisfies this criteria.

17. #34 page 541. Vertices (-5, 0) and (5, 0) and length of minor axis 6.  Find an equation of an ellipse in standard form which satisfies this criteria.

18. #40 page 541 

19. #41 page 541. 3(x + 2)2 + 4(y – 1)2 = 192.   Find the center, major vertices and foci. Sketch the graph.

20. Find the equation of  an ellipse with vertices (0, 4) and (0, -4) and eccentricity ¼.

21. #51 page 541

22.  # 54 page 542

23. #53 page 542

24. #65 page 542

25. #71 page 543