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 $N\to\infty$?