1

$c^2=a^2-b^2$ is used when determining the foci of an ellipsis. However, it is unclear where this formula arises from. It is not at all intuitive for me. A previous answer I found on Stack Exchange was the following:

Solution

enter image description here

Using the diagram above, the formula can be derived by pythagoras. However, I am unsure why that length is $a$. I do not understand there explanation. $a$ is that distance on the $x$ axis, so why is the length shown on the diagram also $a$?

I am looking for a solution that clearly explains the derivation and intuition for this formula used to find foci.

James Chadwick
  • 743
  • 2
    Because the sum of the distances of any point of the ellipse from the foci is $2a$. – Intelligenti pauca Aug 29 '23 at 07:39
  • 1
    An ellipse can be defined as the set of points $M$ of the plance, such that $\mathrm{dist}(M,F_1) + \mathrm{dist}(M,F_2) = 2a = const$. Considering the points equidistant to both foci, i.e. $M = (0,\pm b)$, and the Pythagorean theorem, you'll end up to the wanted relation. – Abezhiko Aug 29 '23 at 07:42
  • It is explained in this answer https://math.stackexchange.com/a/2673890/42969 to the linked question. – Martin R Aug 29 '23 at 07:43

1 Answers1

4

One definition of an ellipse is the geometric shape formed by the set of all points $P$ in the plane so that the sum of distances to the foci, $PF_1+PF_2$ is constant.

Now, if we pick the point $P$ to be the point of the ellipse on the positive $x$ axis, this sum of distances is $(a-c)+(a+c)=2a$. On the other hand, if we pick $P$ to be the point on the positive $y$ axis, the sum of distances is clearly twice of the diagonal length indicated in your diagram in the question. By the property/definition of the ellipse above, this shows that your length is $a$.

YiFan Tey
  • 17,431
  • 4
  • 28
  • 66