Derivation of the cartesian formula for an ellipse

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This mathematics article is devoted entirely to providing mathematical proofs and support for claims and statements made in the article ellipse. This article is currently an experimental vehicle to see how we might be able to provide proofs and details for math articles without cluttering up the main article itself. See [WikiProject Mathematics/Proofs] for current discussion. This article is "experimental" in that it is a proposal for one way that we might be able to deal with expressing proofs.

The derivation of the cartesian form for an ellipse is simple and instructive. An ellipse is defined as a the loci of points equidistant to two fixed points called the foci. Assuming that the foci are located at (-c,0) and (c,0) (ie. the ellipse is centered at (0,0)) then the sum of the distance between any point (x,y) and the two foci is constant.

If (x,y) is any point on the ellipse and if [d_1] is the distance between (x,y) and (-c,0) and [d_2] is the distance between (x,y) and (c,0), i.e.

[d_1 = \sqrt ]

[d_2 = \sqrt ]

then

[d_1 + d_2 = 2a]

where a is the semimajor axis. From this we can derive the cartesian equation. Substituting:

[\sqrt + \sqrt = 2a]

To simplify we isolate the radical and square both sides.

[\sqrt = 2a - \sqrt ]

[(x+c)^2 + y^2 = \left ( 2a - \sqrt \right )^2]

[(x+c)^2 + y^2 = 4a^2 - 4a\sqrt + (x-c)^2 +y^2]

Solving for the root and simplifying:

[\sqrt = - ((x+c)^2+y^2-4a^2-(x-c)^2-y^2) ]

[\sqrt = - (x^2 + 2xc + c^2 -4a^2 -x^2 +2xc -c^2)]

[\sqrt = - (4xc - 4a^2)]

[\sqrt = a - x]

A final squaring

[(x-c)^2+y^2 = a^2 - 2cx + x^2]

[x^2 - 2xc + c^2 + y^2 = a^2 -2xc + x^2]

[x^2 + c^2 + y^2 = a^2 + x^2]

Grouping the x-terms and dividing with [a^2-c^2]

[x^2 \left( 1 - \right) + y^2 = a^2 - c^2]

[x^2 \left( \right) + y^2 = a^2 - c^2]

[ + = 1]

If x = 0 then [d_1 = d_2 = a = \sqrt ]

Therefore we can substitute

[b^2 = a^2-c^2]

And we have our desired equation:

[ + = 1]

Proof that d1 + d2 = 2a http://math.usask.ca/conicsdemo/DEMO/extensions/page2c.html

 

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