# Parameters of an elliptic equation?

I know that an ellipse equation described by:

$\frac {x^2} {a^2} + \frac {y^2} {b^2} =1$

My question is in the equation above how many parameters we need to estimate? Two or four? The unknows parameters are only a, b or are also x, y???

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I think there is no elliptic function here. Please consider retagging it. Thanks. –  awllower Feb 24 '13 at 16:31

For every fixed ellipse, $a$ and $b$ are determined, and $x$ and $y$ are variables. The ellipse itself is the collection of points in the plane whose $x$ and $y$-coordinates statisfy the equation. For instance, take the ellipse $$\frac{x^2}{1} + \frac{y^2}{4} = 1$$ then $a=1$ and $b=2$. For any point $(x_0, y_0)$, if it is the case that $\frac{x_0^2}{1} + \frac{y_0^2}{4} = 1$, then that point is on the ellipse. For any point on the ellipse, if you take the coordinates and put them into the equation, the left-hand side and right-hand side will turn out equal. For any other point in the plane (like the origin, or say $(535, -1234)$), the equation is not statisfied, and therefore they are not on the ellipse.
Any other coice of $a$ and $b$ will result in a different ellipse. So, $a$ and $b$ are numbers that are determined beforehand, like $1$ and $2$ above, or any other pair of numbers, really. The reason they are called $a$ and $b$, and not written out specifically is in order to cover any concievable choice of numbers. This is something done often in mathematics (and quite successfully at that, if you can get your head around it).
The weakness of using letters instead of numbers like this is that you lose the immediate and concrete grasp on the numbers involved if you're not used to it. The strengths, however, is that you can talk about them all at once, instead of taking care of every single choice one by one. Imagine if you want to say something about all those ellipses, and having to go through the same calculations for every possible choice of denominators. Or just call them $a$ and $b$, whatever they are, and do the calculations once. It's pretty clear which approach will be more fruitfull in the long run.