Angle Between a Pair of Straight Lines

For the pair of lines through the origin $ax^2+2hxy+by^2=0$, dividing by $x^2$ and substituting $m=y/x$ gives $bm^2+2hm+a=0$, whose roots $m_1,m_2$ (the two slopes) satisfy, by the sum/product-of-roots relations,

The angle $\theta$ between the two lines then follows from the general angle-between-two-lines formula (§6.4):

(Here $(m_2-m_1)^2=(m_1+m_2)^2-4m_1m_2$ is used to express the difference of roots via their sum and product, without needing $m_1,m_2$ individually.)

As a direct consequence, the pair of lines is:

• real and distinct if $m_1,m_2$ are real and unequal, i.e. $h^2-ab>0$;
• real and coincident (the 'two' lines are actually the same line) if $m_1,m_2$ are real and equal, i.e. $h^2-ab=0$;
• not real (imaginary) if $m_1,m_2$ are not real, i.e. $h^2-ab<0$ (only the origin then satisfies the equation). …