This is the problem from my book I am working on.
Find a value for $A$, $B$ for which $Ax+By+C=0$ is not a line.
I got $0$ and $ \infty $, are there any others?
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If $A \neq 0$ or $B \neq 0$ then the equation is that of a line.
Otherwise, if $A = B = 0$, then there are two cases depending on the value of $C$. If $C=0$, then the equation becomes: $$0=0$$ which is true for all $(x,y) \in \mathbb R^2$; in other words, it describes the $x,y$-plane and not a line. On the other hand, if $C\neq 0$, then the equation is equivalent to $$0=1$$ which is not true for any $(x,y) \in \mathbb R^2$. The empty set is not a line.