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In an ellipse I understand that the terms $a$ and $b$ are used to refer to the lengths of semi major and semi minor axes respectively. In my textbook there are different formulae for each of the cases; when $a>b$ and vice versa. However my teacher has said that it is okay to always take the (semi) length of axis along $x$-axis (I am talking about standard ellipses) as $a$ and correspondingly along $y$ as $b$. Which convention should I follow?

Nate william
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  • See Ellipse. – Mauro ALLEGRANZA Dec 12 '19 at 12:02
  • The major axis is always larger than the minor one. Therefore it is not clear what do you mean by saying " when > and vice versa". If "vice versa" is possible the author of the textbook uses another definition for $a$ and $b$. – user Dec 12 '19 at 12:06
  • I guess a and b are just variables and do not represent semi major and minor axes as such. – Nate william Dec 12 '19 at 12:36
  • Sometimes you don't care which axis is longer; eg, maybe you're interested in the "stretched circle" shape but not where the foci are located. You have to call their lengths something, and it's reasonable to associate $a$ with $x$, and $b$ with $y$, so you have "horizontal/vertical" axes instead of "major/minor". It's not unlike having the quadratic equation $bx^2+cx+a=0$: the Quadratic Formula still applies to find $x$, but it'll look like $\frac{1}{2b}(-c\pm \sqrt{c^2-4ab})$; you have to pay attention to the roles of the coefficients $a$, $b$, $c$, not their alphabetical order. – Blue Dec 12 '19 at 12:44
  • See, as one example among many, this answer of mine, where I specifically refer to "horizontal" and "vertical" semi-axes (although I use "radii", a term I like better). – Blue Dec 12 '19 at 12:51
  • The characterization of an ellipse via major an minor axes is universal. It does not depend on a certain coordinate system. Contradistinctively, the system proposed by your teacher has meaning only in a Cartesian coordinate system with a special choice of the axes. – user Dec 12 '19 at 13:24
  • But if a>b, does that make it necessary that the ellipse is wider along the axis and correspondingly if a<b should the ellipse always be wider along the y axis? And does the same apply to transverse and conjugate hyperbolas respectively? – Nate william Dec 13 '19 at 02:02

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