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Ok, so apparently this question was somewhat asked in When does a conic represent an empty set?, but since it is non-understandable I will ask it anyways.

According to the plethora of articles, class notes etc. I found online, the shape of a conic, described by the equation $$Ax^2+Bxy+Cy^2+Dx+Ey+F=0,$$ can be determined by looking at $$\det\begin{bmatrix}2A& B &D\\B&2C&E\\D&E&2F\end{bmatrix},$$ which can be zero or not, and by the discriminant $B^2-4AC$. This gives 6 possibilities: An ellipse, a parabola, a hyperbola, a point, a pair of parallel lines (possibly the same), and two different non-parallel lines.

However there seems to be a missing possibility: The empty set.

For example, $x^2+1=0$ determines the empty set. If you don't like this equation, take $(x+1)^2+(y+1)^2+(x+y)^2+1=0$, which has all nonzero coefficients but still determines the empty set.

When does a quadratic equation such as the one above determine the empty set?

I have no idea.

311411
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2 Answers2

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There are really two cases for the non-zero determinant. Consider $x^2 + y^2 - 1 = 0$ and $x^2 + y^2 + 1 = 0.$ In one case we have $$\begin{bmatrix}2A& B &D\\B&2C&E\\D&E&2F\end{bmatrix} = \begin{bmatrix}2 & 0 &0\\0&2&0\\0&0&-2\end{bmatrix} $$ and in the other we have $$\begin{bmatrix}2A& B &D\\B&2C&E\\D&E&2F\end{bmatrix} = \begin{bmatrix}2 & 0 &0\\0&2&0\\0&0&2\end{bmatrix}. $$ Non-zero determinants both times, but one case is an ellipse (in fact, a circle) and the other is the empty set.

Moreover, $x^2 - 1 = 0$ leads to a zero determinant (the only non-zero entries are $2A$ and $2F$) and it is a pair of parallel lines, but if we just change $F$ from $-1$ to $1$ we get the empty set you exhibited.

I tried to tease apart the cases in a slightly different way in https://math.stackexchange.com/a/2096865 -- in summary, for non-zero discriminant one can transform the equation into one that is symmetric around the origin (and then the sign of the constant term tells whether the solution set is empty), and for the zero discriminant there is a case that reduces to a quadratic equation in some combination of $x$ and $y$, and the discriminant of that equation tells you whether the solution is two parallel lines, one line (the two parallel lines merged into one), or empty.

David K
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There are many cases, in fact most often a randomly cjosen quadratic of $x$ and $y$ $$ax^2+by^2+2hxy+2gx+2fy+c=0~~~(1)$$ does so for instance: $x^2+y^2+1=0.$ No other method can be better than seeing (1) as a quadratic of $x$ and writing $$ ax^2+x(2g+2hy)+by^2+2fy+c=0.$$ Then $$x=\frac{-(g+hy)\pm \sqrt{(g+hy)^2-a(by^2+2fy+c)}}{a}.$$ Next the discriminant under square-root tells everything. For instance if it is $$-7(y^2+y+1), -3y^2, -5~\mbox{or}~ -4y^2-2,$$ then $x$ is non-real so no curve can exist in $(x,y)$ plane and it is called empty or null set.

Z Ahmed
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