Conics

[just2011]

"The foci of an ellipse are two special points F₁ and F₂ on the ellipse's major axis and are equidistant from the center point. The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis ( PF₁ + PF₂ = 2a ). Each of these two points is called a focus of the ellipse."
From Wikipedia.
how would I show the major axis is 2a?

 

[HallsofIvy]

"The foci of an ellipse are two special points F₁ and F₂ on the ellipse's major axis and are equidistant from the center point. The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis ( PF₁ + PF₂ = 2a ). Each of these two points is called a focus of the ellipse."
From Wikipedia.
how would I show the major axis is 2a?

I'm not sure what you mean by this. In the quote you are told that the major axis is 2a. If that, what do you mean by "a".

If you are asking "how do I show that PF1+ PF2 is equal to the length of the major axis", then let P be one of the two points where the major axis crosses the ellipse- specifically, the one closer to F1. Call the length of the major axis "L" (2a above) and let f be the distance from the center of the ellipse to F1. Then PF1= L/2- f- the distance form P to the center of the ellipse, minus the distance from F1 to the center. Because of symmetry, the distance from the center of the ellipse to F2 is also f so the distance from P to F2 is the distance from P to the center, L/2, plus the distance from the center to F2: PF2= L/2+ f. So PF1+ PF2= (L/2- f)+ (L/2+ f)= L, length of the major axis and, so 2a where a is defined as the length of the "major semi-axes", the distance form the center of the ellipse to P.

 

[soroban]

Hello, just2011!

The foci of an ellipse are two special points F1 and F2 on the ellipse's major axis
and are equidistant from the center points. The sum of the distances of any P on
the ellipse to the two foci is constant and equal to the major axis (PF1 + PF2 = 2a).
Each of these two points is called a focus of the ellipse.

How would I show the major axis is 2a?

Maybe a sketch?

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              *   *   *
          *       |       *
        *         |         *
       *          |          *
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      *           |           *
  - - o - - o - - + - - o - - o - -
    -a*    F2     |    F1     *a
                  | 
       *          |          *
        *         |         *
          *       |       *
              *   *   *
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