What is an imaginary line?

Question

An imaginary point is defined as an ordered pair of values, at least one of which is complex.
My text says that if $h^2 < ab$ in the equation : $$ ax^2 + 2hxy + by^2 $$ it represents two imaginary lines, which intersect in a real point. According to Wikipedia, an imaginary curve is one which does not contain any real points.
My questions are, what is an imaginary line, how does an imaginary line differ from any line on the complex plane, how is it possible that an imaginary line can have a real point, and how can it be proven that the two imaginary lines described by the equation meet in a real point?

Answer 1

In this case I think "imaginary line" is intended to mean a 1-dimensional $\mathbf C$-linear subspace of $\mathbf C^2$ whose intersection with $\mathbf R^2$ is just the point $(0,0)$. So to answer your questions in order:

1. In particular, it is not a line in the complex plane, if "complex plane" has its usual meaning, namely $\mathbf C$.

2. It is possible simply because Wikipedia's definition of "imaginary curve" is not consistent with the one the author in your case has in mind.

3. To see how the lines intersect, factor $ax^2+2hxy+by^2$ into linear factors. (Exercise: show you can always do this.) You get an expression that looks like

$$(\alpha x + \beta y)(\gamma x + \delta y)$$ for some complex numbers $\alpha,\beta,\gamma,\delta$. The two lines referred to are then the sets of points $(x,y) \in \mathbf C^2$ where each of these two linear factors takes the value zero. It is clear that both lines then contain the point $(0,0)$; moreover, as long as the lines are distinct, which will be true as long as $\alpha\delta \neq \beta \gamma$, then (exercise!) they won't intersect in any other points.