Thread: Proof that the sum of the distances from any point on a ellipse to foci is always 2a

1. Proof that the sum of the distances from any point on a ellipse to foci is always 2a

Hi, can someone please tell me the proof of the following statement:

For the ellipse
$\displaystyle x^2/a^2 + y^2/b^2 = 1$
the distances from any point on the ellipse to the foci [which are at (ae,0) and (-ae,0)] always total up to 2a.

Thanks.

2. Originally Posted by AAKhan07
Hi, can someone please tell me the proof of the following statement:

For the ellipse
$\displaystyle x^2/a^2 + y^2/b^2 = 1$
the distances from any point on the ellipse to the foci [which are at (ae,0) and (-ae,0)] always total up to 2a.

Thanks.

3. Originally Posted by AAKhan07
Hi, can someone please tell me the proof of the following statement:

For the ellipse
$\displaystyle x^2/a^2 + y^2/b^2 = 1$
the distances from any point on the ellipse to the foci [which are at (ae,0) and (-ae,0)] always total up to 2a.

Thanks.
That's an interesting question. Typically, an ellipse is defined as the figure such that the sum of distances from a point on the figure to the foci is constant and then the equation is derived from that. Here you are doing it the other way around.

The sum of the two distance is, of course, $\displaystyle \sqrt{(x-ae)^2+ y^2}+ \sqrt{(x+ ae)^2+ y^2}$. I would recommend that you do "top" and "bottom" separately. That is, first replace y by $\displaystyle y= b\sqrt{1-\frac{x^2}{a^2}}$ and then by $\displaystyle y= -b\sqrt{1-\frac{x^2}{a^2}}$. Of course, since y is squared thos will give the same thing.-

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proof of sum of distances from both foci of an ellipse is equal to 2a

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