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

 A: There are two common, but distinct, conventions here.

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).

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.

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.
