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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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In my opinion, $A^2+B^2 \neq 0$ is the only condition. – Siminore Aug 29 '12 at 12:33
@Siminore could you give me a reason why? Or at least point me in the correct direction – yiyi Aug 29 '12 at 12:40
What is the book's definition of "a line"? $A^2+B^2=0\implies A=0$ and $B=0$, so if we have $C\ne0$ the equation is inconsistent, which is probably what @Siminore was thinking. Considering $\infty$ is not really appropriate here, I don't think many books would call $\infty$ a "value" (this terminology, and the fact that this is tagged as algebra-precalculus, suggests that "value" means "real number"). – process91 Aug 29 '12 at 12:42
@MichaelBoratko its for A-levels so complex numbers should be allowed. The book just lists some various equations of lines and then has some problems. – yiyi Aug 29 '12 at 12:46
@MaoYiyi If complex numbers are allowed then are $x,y\in\Bbb C$? If so, then many values don't represent lines in $\Bbb C$... – process91 Aug 29 '12 at 12:48
up vote 5 down vote accepted

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.

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