In the ellipse-related formula $c^2=a^2-b^2$, is $a$ always the biggest number, or can it be the smallest?

Question

Referring to Ellipses:

In the formula $c^2= a^2-b^2$, is the $a$ always the biggest number or can it be the smallest?

Answer 1

There are two common, but distinct, conventions here.

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One convention is that, whatever the length of the semi-major axis is, we call that $a$ (and the semi-minor axis $b$). The semi-major axis is the half the larger of the two axes of an ellipse, as seen in the following image from Wikipedia:

In an ellipse, the semi-major axis, $a$, is longer than the semi-minor axis, $b$, so under this convention $a$ is always the larger of the two numbers (if we define $a$ and $b$ this way, as lengths, we are assured both are positive).

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The other convention is that we arbitrarily call the semi-major axis $a$ or $b$; in this situation, $a$ could be smaller than $b$. Or it could be bigger. We just don't know in advance.

In this case, we cannot use the equation $c^2 = a^2 - b^2$, because $|b|$ may well be larger, in which case $a^2 - b^2$ would be negative (but $c^2$ can't possibly be negative). So that equation for the focal length $c$ needs to be modified; instead we use $$c^2 = |a^2 - b^2|,$$ where the absolute value forces $|a^2 - b^2|$ to be positive, and all is well.

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If your teacher/textbook are careful, and you see the equation $c^2 = a^2 - b^2$, then they must be following the first convention in which $|a|$ is larger than $|b|$.

But if you see the equation $c^2 = |a^2 - b^2|$, they are following the second convention and you don't know which of $|a|$ or $|b|$ is larger, in advance.