Aman Bansal
Last Activity: 12 Years ago
Dear Simhadri,
The derivation of the cartesian form for an ellipse is simple and instructive. One simple definition of the ellipse is the "locus of all points of the plane whose distances to two fixed points(called the foci) add to the same constant". See ellipse for other definitions.
Let the foci be points (-c,0) and (c,0). Then the ellipse center is (0, 0).If (x,y) is any point on the ellipse and if
is the distance between (x,y) and (-c,0) and
is the distance between (x,y) and (c,0), i.e.

then we can define 

(a here is the semi-major axis, although this is irrelevant for the sake of the proof). From this simple definition we can derive the cartesian equation. Substituting:

To simplify we isolate the radical and square both sides.



Solving for the root and simplifying:


Swap sides to return to original format and continue:




A final squaring



Grouping the x-terms and dividing by 


Where: 


If x = 0 then

(where b is the semi-minor axis)
Therefore we can substitute

And we have our desired equation:

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Thanks
Aman Bansal
Askiitian Expert