The Ellipse

Definition and equation

Take two points F and F' and a strictly positive value 2a such that 2a is greater than |F,F'|.
The locus of all points D such that |D,F| + |D,F'| = 2a is an ellipse.
We choose the line FF' as x-axis and the perpendicular bissector of the segment [F,F'] as y-axis.
We give F and F' resp. coordinates (c,0) and (-c,0).

We see that a>c
The points F and F' are called the foci of the ellipse 

 
        D(x,y) is on the ellipse
                <=>
        |D,F| + |D,F'| =  2a
                <=>
  _______________      _______________
 |        2    2      |        2    2
\| (x - c)  + y   +  \| (x + c)  + y    = 2a

Squaring        <=>
                                    ________________________________
       2    2          2    2      |         2    2          2    2      2
(x - c)  + y  + (x + c)  + y  + 2 \| ((x + c)  + y ) ((x - c)  + y ) = 4a

                <=>
                ...

                <=>
  ________________________________
 |         2    2          2    2
\| ((x + c)  + y ) ((x - c)  + y ) = 2 a2  - (x2  + y2  + c2 )

                <=>

  _______________________________________________
 |   2    2    2            2    2    2
\| (x  + c  + y  + 2 c x) (x  + c  + y  - 2 c x) = 2 a2  - (x2  + y2  + c2 )   (*)

Squaring        =>


(x2  + y2  + c2 )2 - 4 x2 c2 = 4 a4  - 4 a2  (x2  + y2  + c2 ) + (x2  + y2  + c2 )2    (**)

                <=>

        - 4 x2  c2 = 4 a4  - 4 a2  (x2  + y2  + c2 )
                <=>
                ...

                <=>

        (a 2 - c2 ) x2  + a 2 y2  = a2  (a2  - c2 )

                       Since a > c , we can say  a2  - c2  = b 2

                <=>

        b2 x2  + a2  y2  = a2  b2

                <=>
           2    2
          x    y
          -- + -- = 1           (***)
           2    2
          a    b
This is the equation of the ellipse.
But we don't know if (*) and (**) are equivalent because we don't know if the right side of (*) is a positive value.
From above we know that if |D,F| + |D,F'| = 2a then (***) holds.
To prove the reverse, it is sufficient to show that if (***) holds, then the right side of (*) is a positive value.
Well, from (***) we have 
 

        x2                 y2
        -- < or = 1  and    -- < or = 1
        a2                 b2


then    x2< or = a2   and y2 < or = b2


then    x2 + y2  < or = a2  + b2


then    x2  + y2  + c2  <  or = a2  + b2 + c2


then    x2  + y2  + c2  <  or =   2a2     since  b2 + c2  = a2

then the right side of (*) is a positive value.
Now |D,F| + |D,F'| = 2a and (***) are equivalent.
The intersection points of the ellipse with the x-axis are A'(-a,0) and A(a,0). These are the vertices on the x-axis.
The intersection points of the ellipse with the y-axis are B'(-b,0) and B(b,0). These are the vertices on the y-axis.
The segment [A',A] is called major axis of the ellipse.
The segment [B',B] is called minor axis of the ellipse.
The segments [D,F'] and [D,F] are the focal radii through point D.

the circle as a special ellipse

If F = F' then c = 0 and a = b and then |D,F| + |D,F| = 2a
Hence |D,F| = a .
The ellipse is a circle with equation 
 
x2  + y2  = a2
The radius is a.

Parametric equations of the ellipse

Take in a plane two lines l and m with resp. equations 
 
        x = a cos(t)            (1)
        y = b sin(t)            (2)
The real numer t is the parameter.
We know, from the theory of 'Elimination of parameters', that the intersection points of the two associated lines constitute a curve. To obtain the equation of that curve, we eliminate the parameter t from the two equations. This means that we search for the condition such that (1) and (2) has a solution for t.
The simultaneous equations (1) and (2) are equivalent to 
 
        x / a = cos(t)
        y / b = sin(t)
This system has a solution for t if and only if 
 

        sin2 (t) + cos2 (t) = 1
<=>
          x2    y2
          -- +  -- = 1
          a2    b2
Hence, the two associated lines constitute a curve and that curve is the ellipse.
We say that (1) and (2) are parametric equations of the ellipse.
The point 
 
        D(a cos(t) , b sin(t))
is on the ellipse for each t-value and with each point of the ellipse corresponds a t-value. From this it follows, as a special case, that 
 
        x = a cos(t)
        y = a sin(t)
are parametric equations of the circle with radius a.
Then, the point D(a cos(t) , a sin(t)) is a variable point of that circle.

Another approuch of the ellipse

Again, take a variable point D(a cos(t) , a sin(t)) of the circle with radius a.
Now, we compress the circle in the y-direction with a factor b/a. The coordinates of the variable point of the new curve are D(a cos(t) , b sin(t)).
From this, we see that the new curve is the ellipse 
 
          x2    y2
          -- +  -- = 1
          a2    b2

Tangent line in a point D of an ellipse

Take the ellipse 
 
          x2    y2
          -- +  -- = 1
          a2    b2
To obtain the slope of the tangent line we differentiate implicitly. 
 
          2x   2y y'
          -- + ----- = 0
          a2    b2
Solving for y', we obtain 
 
             b2  x
       y'= - ----
             a2  y
Say D(xo,yo) is a fixed point of the ellipse.
The slope of the tangent line in point D is 
 
             b2 xo
       y'= - ------
             a2 yo
The equation of the tangent line is 
 
                 b2 xo
      y - yo = - ----- (x - xo)
                 a2 yo
<=>
       a2  yo y - a2  yo2  = b2  xo2  - b2  xo x
<=>
        a2  yo y + b2  xo x = a2 yo2  + b2  xo
<=>
                since D(xo,yo) is on the ellipse
        a2  yo y + b2  xo x = a2 b2
<=>

         xo x   yo y
         ---- + ---- = 1
          a2     b2
The last equation is the tangent line in point D(xo,yo) of an ellipse.

Tangent line as bisecting line

Take the bisectors t and n of the lines DF and DF'.
Say F" is the reflection point of F with respect to t.
Take any point T on t different from D.

Since |D,F| = |D,F"| , |F',F"| = 2a .
Now in the triangle F'TF" , we see that 

 
        |F',T| + |T,F"| > 2a

=>      |T,F'| + |T,F| > 2a
And from the definition of ellipse, it follows that T is outside of the ellipse. Hence all the points of t, different from D, are outside of the ellipse and therefore the bissector t of the lines DF and DF' is a tangent line of the ellipse.
The line n is a normal of the ellipse.

Properties

Since |F',F"| = 2a = constant, we see that the mirror image of F with respect to a variable tangent line is on the circle with center F' and with radius 2a.
Call P the projection of F on the tangent line.
Point O is the midpoint of the segment [F,F'] and point P is midpoint of the segment [F,F"]. Hence |O,P| = a .
The orthogonal projection of F on a variable tangent line is the circle with center O and radius a.


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