The Ellipse




Definition and equation

We start with a circle C with radius a.
x2+ y2 = a2
If we multiply the y-value of all points P' on the circle by a fixed value b/a, then each point P' is transformed in a new point P.

The set of all these points P is an ellipse.

 
   P(x,y) is on the ellipse
<=>
   P'(x, (a/b) y ) is on  C
<=>
   x2  + (a/b)2 y2 = a2
<=>
   x2     y2
  ---- + ----  = 1
   a2     b2
The points A(a,0) A'(-a,0) B(0,b) B'(0,-b) are called the vertices of the ellipse.
[AA'] is called the major axis and has 2a as length.
[BB'] is called the minor axis and has 2b as length.

If a = b then the ellipse is a circle.

Parametric equations of the ellipse

On the previous figure we see that P'(a cos(t) , a sin(t))
By the definition of P we have P(a cos(t) , b sin(t))
From this it follows that
 
    x = a cos(t)
    y = b sin(t)
are parametric equations of the ellipse.

One can construct, for every t, a point on the ellipse.

Draw a line parallel to the y-axis through the intersection point of OP' and the circle with radius a.
Draw a line parallel to the x-axis through the intersection point of OP' and the circle with radius b.
The intersection point of these lines is a point of the ellipse.

Basic geometric property

We define c as c2 = a2 - b2 met c > 0 or c = 0
We define the points F(c,0) and F'(-c,0) as the foci of the ellipse.
Note that the distance from a focus to point B(0,b) is equal to a.

Say P(a cos(t) , b sin(t)) is a variable point of the ellipse.

 
  |PF|2 = (a cos(t) - c)2 + b2 sin2(t)

         =  a2 cos2(t) - 2 a c cos(t) + c2 + (a2 - c2) sin2(t)

         =  a2 - 2 a c cos(t) + c2 cos2(t)

         =  (a - c cos(t))2

   |PF'|2 = ... = (a + c cos(t))2

Since  a > c :

   |PF| + |PF'| = a - c cos(t) + a + c cos(t) = 2a = constant
Conversely, we show that point P is on the ellipse if |PF| + |PF'| = 2a.

Connect P with F and F'. Say P' in the intersection point of F'P with the ellipse. (see figure)

If P is not on the ellipse then

 
       |PF| + |PF'| = 2a
 and   |P'F| + |P'F'| = 2a

We have |PF| + |PF'| -|P'F| - |P'F'| = 0
Then, we have |P'P| + |PF| = |P'F| and this is impossible.

Conclusion:

If F and F' are the foci of een ellipse x2/a2 + y2/b2 = 1 , then we have for each point P of the ellipse |PF| + |PF'| = 2a.

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  xo2
<=>
                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 in 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 bisector 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.

Solved problems

The given solution is not 'the' solution.
Most exercises can be solved in different ways.
It is strongly recommended that you at least try to solve the problem before you read the solution.

Normal line

Find the equation of the normal line in a point P(xo,yo) of the ellipse
 
        b2 x2 + a2 y2 = a2 b2

Concurrent lines

Take on the ellipse E a variable point P and F is F(c,0).
Show that the following lines are concurrent.
  • The tangent line to the ellipse in P
  • The perpendicular to PF in F
  • The line x = a2/c (directrix associated to F)

Eccentricity

P is a variable point on the ellipse E and F is F(c,0).
d is the line x = a2/c (directrix associated to F).
Show that |PF|/|P,d| is constant.

The distances |PF| and |PF'|

P( a cos(t), b sin(t) ) is on the ellipse E.
Calculate the distances |PF| en |PF'|

Foci and tangent Line

Find the product of the distances from the foci of an ellipse to a fixed tangent line. Show that this product is constant.

Midpoints of parallel chords

In an ellipse E take all chords with slope m. Show that the centers of these chords are on one line.

Locus

Line r has an equation 3x - 4y = 0
Line r'has an equation 3x + 4y = 0
Find the locus of the points P such that
| P,r |2 + | P,r' |2 = 10

Two right angled triangles

P is a variable point of the ellipse b2 x2 + a2 y2 =a2 b2.
Let A = A(a,0) and A' = A'(-a,0).
M1 is on the x-axis such that APM1 is a right angled triangle in P.
M2 is on the x-axis such that A'PM2 is a right angled triangle in P.
Show that the distance |M1 M2| is constant.

Tangent line and minimum

A variable tangent line to the standard ellipse forms with the x-axis and the y-axis a triangle. Find the tangent lines such that the area of the triangle is minimum.




Topics and Problems

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