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Given an arbitrary conic section in the form $$Ax^2 + Bxy + Cy^2 + Dx + Ey +F=0$$ (Where the coefficients are real valued) is there a simple test which can determine whether or not a particular conic is an ellipse? I know that if a conic section is an ellipse, then $A$ and $C$ will have the same sign, however I am not sure if this is a sufficient condition as well.

Edit: Forgot to include the "$...+F=0$".

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    $\begingroup$ See en.wikipedia.org/wiki/… $\endgroup$ – lab bhattacharjee Feb 5 '18 at 5:31
  • $\begingroup$ @lab bhattacharjee from the link it says to evaluate $B^2 - 4AC$ - do I need to rotate the conic section first though? $\endgroup$ – Jbag1212 Feb 5 '18 at 5:37
  • $\begingroup$ Just check if $B^2-4AC<0$ $\endgroup$ – lab bhattacharjee Feb 5 '18 at 5:52
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    $\begingroup$ No, the quantity $B^2-4AC$ is invariant under rotations. You might want to check this: it’s messy, but very satisfying to come out with the result. $\endgroup$ – Lubin Feb 5 '18 at 5:52
  • $\begingroup$ Your ellipse necessarily passes through the origin? $\endgroup$ – Bernard Feb 5 '18 at 9:15
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Here is the answer for the normalised equation of a non-degenerate conic $$Ax^2+2Bxy+Cy^2+2Dx+2Ey+T=0.$$

Consider the matrix $$M=\begin{bmatrix} A & B & D \\ B & C &E \\ D &E & F \end{bmatrix} $$ and the matrix of the quadratic part of the equation $$Q=\begin{bmatrix} A & B \\ B & C \end{bmatrix}. $$ The conic is non-degenerate if and only if $\det M\ne 0$. Further, the conic is an ellipse if and only if:

  1. The quadratic part of the equation (associated to the matrix $Q$) has signature $(2,0)$;
  2. The quadratic form on $\mathbf R^3$ associated to $M$ has signature $(2,1)$.
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  • $\begingroup$ Is there an advantage to using this method over checking $B^2 - 4AC$? Does checking $B^2 - 4AC$ only work if it "passes through the origin"? $\endgroup$ – Jbag1212 Feb 5 '18 at 21:46
  • $\begingroup$ The sign $B^2-4AC$ tests the possibility of points at infinity, i.e. the possibility of a hyperbola (2 points at infinity) or a parabola (az double ipoint at infinity. It's equivalent to the signature of $A$, but there may be degenerate cases which are handled by the study of the matrix $M$. In the ellipse case, the curve may be empty or reduced to a point. $\endgroup$ – Bernard Feb 5 '18 at 22:01
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If $ I_2=B^2-4AC$ is negative then it is an ellipse whether or not (RHS being zero ) it passes through origin as special case.

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