Ellipse

Definition:

An ellipse is the set of points for each of which the sum of the distances from two fixed points (foci) is a given constant.  The distance from a focus to a point on the ellipse is called a focal radius.  The point bisecting the line segment joining the focal points is called the center of the ellipse.

To measure the ovalness of an ellipse, the eccentricity is figured.
e = c/a, where c is the distance from the center to the foci, and a is the distance from the center to the vertices on the major axis.

Derive the equation of an ellipse:

Let the center of the ellipse be (0,0), the foci of the ellipse be (c,0) and (-c,0) and the constant sum of the focal radii be 2a where a > c.  (x,y) is a point on the ellipse.
According to the definition,  \/((x + c)2 + (y - 0)2)  +  \/((x - c)2 + (y - 0)2) = 2a
Simplifying the radicands,    \/(x2 + 2cx + c2 + y2)  +  \/(x2 - 2cx + c2 + y2 = 2a
Subtract one radicand,        \/(x2 + 2cx + c2 + y2) = 2a - \/(x2 - 2cx + c2 + y2)
Square both sides,             x2 + 2cx + c2 + y2 = 4a2 - 4a \/(x2 - 2cx + c2 + y2) + x2 - 2cx + c2 + y2
Combine like terms,            4cx = 4a2 - 4a \/(x2 - 2cx + c2 + y2)
Subtract 4a2,                    4cx - 4a2 = - 4a \/(x2 - 2cx + c2 + y2)
Divide by - 4a,                 - cx+ a = \/(x2 - 2cx + c2 + y2)
                                             a
Get common denominator,  -cx + a2  =  \/(x2 - 2cx + c2 + y2)
                                                a
Square both sides,             a4 - 2a2cx + c2x2   =   x2 - 2cx + c2 + y2
                                                                        a2
Multiply by a2,                 a4 - 2a2cx + c2x2   =   a2x2 - 2a2cx + a2c2 + a2y2
Combine like terms,           c2x2 - a2x2 - a2y2 =  a2c2 - a4
Factor,                              (c2 - a2)x2 - a2y2 = a2(c2 - a2)
Divide by a2(c2 - a2),             x2 -      y2      =   1
                                              a2    c2 - a2
Since a > c, reverse the denominator,  x2   +       y2      =   1
                                                          a2         c2 - a2
Let b2 = a2 - c2, and                          x2 +   y2   =   1
                                                           a2      b2

Things to remember about  x2 +   y2   =   1, a > b
                                       a2      b2
1.  x-intercepts are (a,0) and (-a,0)
2.  y-intercepts are (0,b) and (0,-b)
3.  a2 = b2 + c2
4.  Foci are (c,0) and (-c,0)
5.  Sum of focal radii is 2a
6.  The major axis is on the x-axis (horizontal)
7.  Center is (0,0)

Things to remember about  x2 +   y2   =   1, b > a
                                       a2      b2
1.  x-intercepts are (a,0) and (-a,0)
2.  y-intercepts are (0,b) and (0,-b)
3.  b2 = a2 + c2
4.  Foci are (c,0) and (-c,0)
5.  Sum of focal radii is 2a
6.  The major axis is on the y-axis (vertical)
7.  Center is (0,0)

       (x - h)2 +   (y- k)2   =   1    follows the same patterns but the center is now (h,k), the intercepts are (h + a, k),
          a2             b2                                  and foci are (h + c, k) if a > b.  The intercepts are (h, k + a), and foci are
                                                        (h, k + c) if b > a.  The intercepts and foci are measured from the center.

Examples:
1.  Find the standard equation for the ellipse having foci at (1,2) and (5,2) and a major axis of length 4.

     The center is the midpoint of the foci: (3,2)
     The distance from the center to the foci is 2, so c = 2.
     The major axis is 4, meaning the sum of the radii is 4, so a = 4.

2.  Graph  x2 + 2y2 + 4x - 16y - 5 = 0

     Complete the square so the center and intercepts can be seen.
      x2 + 4x + 4y2  - 16y  =5
      x2 + 4x + 4(y2  - 4y)  = 5
      (x2 + 4x + 4) + 4(y2  - 4y + 4)  = 5 + 4 + 16  (Remember to multiply the second 4 by 2 since it is factored)
      (x + 2)2 + 4(y - 2)2 = 25  (Divide by 25)
      (x + 2)2 + (y - 2)2 =  1      (Invert the 4/25 so it is completely in the denominator)
          25          4/25

      This is a horizontal ellipse with center (-2,2) and vertices (0,2), (-4,2), (-2,3), and (-2,1).  Plot the vertices and connect
      them.

3.  Find the center, vertices, foci, and eccentricity of the ellipse in #2.
     The center is  (-2,2), the vertices are (0,2), (-4,2), (-2,3), and (-2,1).
     Since a2 = b2 + c2 ,                                   eccentricity = c/a = (\/(621/25)) / 5
            25 = 4/25 + c2
              25 - 4/25 = c2
            625 - 4    =  c2
                  25
              621   =   c2
                 25
              \/(621/25) = c
       So the foci are (-2 + \/(621/25), 2)

Problems (there are no problems here yet)

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