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In Keith Kendig's paper, Stalking the Wild Ellipse (published in the American Mathematical Monthly, November 1995), he says that if $A, B, C$ are chosen at random, the probability that the Cartesian equation $Ax^2+Bxy+Cy^2 = 1$ defines an ellipse is about $0.19$. How does one make this precise?

I assume that this statement is similar to, for example, the idea that the probability that "two random integers" are relatively prime is $\frac{6}{\pi^2}$. We choose uniformly from the range $1$ to $N$ and then look at the limit as $N\to\infty$.

So for the ellipse problem, do we choose $A,B,C$ from an $N \times N \times N$ cube (centered around the origin) and then compute a triple integral with $N\to\infty$?

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    It is not clear to me that one should prefer a cube to, say, a sphere, and it is not clear to me that you'll get the same answer with both.2012-05-02
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    Thanks Qiaochu, this is the nut of my question, which I perhaps didn't explain well.2012-05-02

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