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I was reading my textbook where it says, the general equation of second degree $ax^2 + 2hxy + by^2 + 2gx + 2fy +c=0$ represents a pair of straight lines if $abc+2fgh-af^2-bg^2-ch^2 =0$

Then i was also able to know that,

The angle between the lines is $tan^{-1}\frac{2\sqrt{h^2 -ab}}{a+b}$

If $h^2 -ab<0$ then what happens? Will it still represent a pair of straight lines when $abc+2fgh-af^2-bg^2-ch^2 =0$ but $h^2 -ab<0 ?$

Is $abc+2fgh-af^2-bg^2-ch^2 =0$ the necessary condition for representing a pair of straight lines? Or there's something else i don't know of?

Tasneem Bhuiyan
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    This equation describes a conical section. The conical section is a pair of straight lines when the secant plain goes through the vertex. When the discriminant is negative, the plain does not cut the cone at all, and the section degenerates to a single point. – user58697 Jan 22 '24 at 23:14
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    You can add a carefully chosen constant to the equation of a real ellipse or hyperbola to get a point or a pair of lines. The idea is that the rank of the quadratic can then drop. Rank one (think $x^2=0$) a double line. Two (think $x^2+y^2=0$) a pair of lines. Three (think $x^2+y^2+z^2=0$) a non-degenerate conic over ${\Bbb P}^2_{\Bbb C}.$ For the affine plane over the real numbers, you’re looking at differences between $x^2+y^2=0$ (a pair of complex lines meeting in $(0,0)$) and $(x+y)(x-y)=0,$ and $x^2+y^2+1=0, x^2+y^2-1=0, x^2-y^2-1=0$ (empty, circle, hyperbola). No to mention the parabola… – Jan-Magnus Økland Jan 22 '24 at 23:20

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