The Ellipse

1
The Ellipse

• Analytic Geometry
• Section 3.3

2
Definition of ellipse

• An ellipse is the set of all points in a plane
  such that the distance from two fixed points
  (foci) on the plane is a constant.

3
Equation of the Ellipse

• The equation of an ellipse with its center at the
  origin has one of two forms

• The position of the a2 (under the x or y) tells
  you whether the horizontal or the vertical axis
  is the major axis of the ellipse.

4
Ellipse

• This ellipse has a horizontal major axis that is
  16 units long.

5
Ellipse

• The minor axis of this ellipse is 10 units in
  length.

6
Foci

• The two foci for this ellipse are the two points
  lying on the horizontal axis that appear to be a
  little over 6 units from the origin. The origin
  is the center of the ellipse. The distance from
  the center to a focus is c.

7

• The segments drawn from the two foci to the point
  (0,5) on the ellipse are each 8 units in length.
  Their total length is 16 units. This total
  length is also the length of the major axis.

8

• Two more segments are added, drawn from the foci
  to the point (2,4.84) on the ellipse. Their
  lengths are 9.556 and 6.434. The sum of these
  lengths is again 16 units.

9

• The two latest segments, drawn to the point
  (7,-2.42) on the ellipse, are 13.463 units and
  2.537 units in length, a sum of 16 units.

10
The Ellipse

• The ends of the major axis are at (a,0) and
  (-a,0).
• The ends of the minor axis are at (0,b) and
  (0,-b).
• The foci are at (c,0) and (-c,0).

11
The Ellipse

• The sum of the distances from point P to the foci
  is 2a.
• Also,

12

• A chord through a focus and perpendicular to the
  major axis is called a latus rectum.

• The endpoints of the two latus recti are found
  using the equivalence

13
Latus Rectum

• When the equation of the ellipse is

So the endpoints of the latus recti are
14
The Ellipse

• The major axis is the vertical axis with
  endpoints (0,13) and (0,-13). The endpoints of
  the major axis are called the vertices. The minor
  axis has endpoints of (5,0) and (-5,0).

15

• The foci are found using
• so the values of c are 12 and -12. The
  coordinates of the foci are (0,12) and (0,-12).

16

• The endpoints of the latus recti are

17
Problem Determining an equation

• Find the equation of the ellipse with foci at
  (8,0) and (-8,0) and a vertex at (12,0).
• First, place these points on axes.
• The F and F are the foci.

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Finding the equation of the ellipse with foci at
(8,0) and (-8,0) and a vertex at (12,0).

• Since the vertex is on the horizontal axis, the
  ellipse will be of the form
• The values of a and b need to be determined.

19
Finding the equation of the ellipse with foci at
(8,0) and (-8,0) and a vertex at (12,0).

• If the foci are at 8 and -8, then c 8. Since a
  vertex is at (12,0), that means that a 12.
• Relating these values to the standard form for
  an ellipse whose center is at the origin and
  whose major axis is horizontal, ,
  
• and the equivalence
  
• applies. Solve for b2 to get
  
• In this case,

20
Finding the equation of the ellipse with foci at
(8,0) and (-8,0) and a vertex at (12,0).

• Since
• The value of a is 12, and a2 is 144.
• The value of b is and b2 is 80.
• So the equation of the ellipse is
  
  
• or

21
Ellipse with center at (h,k)

• The ellipses with their centers at the origin are
  just special cases of the more general ellipse
  with its center at the point (h,k). This more
  general ellipse has a standard formula of

22
Problem Write the equation in standard form

• The general form of the equation is
• After writing this in standard form, also find
  the coordinates of the center, the foci, the ends
  of the major and minor axes, and the ends of each
  latus rectum.

23
Write in standard
form

• First, group the terms with xs and the terms
  with ys, and move the constant to the other side
  of the equation.

24
Write in standard
form

• Now factor out the coefficient of each squared
  term.
• Then complete the square for each variable.

25
To finish the problem

• Simplify on the right.
• Then divide each side by 32.

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The Ellipse

• This ellipse has a center at .
• The major axis is in length, and the
  minor axis is 4 in length, so their endpoints are
  ( ,0) and (- ,0), (0,2) and (0,-2).
• The foci are at (2,0) and (-2,0).

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To finish

• Since the foci are at (2,0) and (-2,0), the
  endpoints of the latus recti are at

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Site and Assignment

• Theres a neat website that you might want to
  look at for more on the ellipse. Its at
  http//mathworld.wolfram.com/Ellipse.html
• Your assignment, due Monday, is
• 3.3 2, 3, 15, 16, 17, 22, 25, 44