Example of Finding the Angle Between the Lines Represented by the Second Degree Homogeneous Equation

Find the lines represented by the second degree homogeneous equation 3{x^2} + 7xy + 2{y^2} = 0. Also find the measure of the angle between them.

We have the second degree homogeneous equation:

3{x^2} + 7xy + 2{y^2} = 0

Calculating the factors of the given equation:

\begin{gathered} 3{x^2} + 6xy + xy + 2{y^2} = 0 \\ \Rightarrow 3x\left( {x + 2y} \right) + y\left( {x + 2y} \right) = 0 \\ \Rightarrow \left( {3x + y} \right)\left( {x + 2y} \right) = 0 \\ \Rightarrow 3x + y = 0,\,\,\,\, \Rightarrow x + 2y = 0 \\ \end{gathered}

This represents the required equations of straight lines passing through the origin.

Now let \theta be the required angle between this pair of lines, and we have

Compare with the general equation of a homogeneous equation a{x^2} + 2hxy + b{y^2} = 0.

Here a = 3,\,\,b = 2,\,\,h = \frac{7}{2}. If \theta is the angle between the pair of lines, then

\begin{gathered} \tan \theta = \frac{{2\sqrt {{h^2} - ab} }}{{a + b}} = \frac{{2\sqrt {\frac{{49}}{4} - 6} }}{{3 + 2}} \\ \Rightarrow \tan \theta = \frac{{\sqrt {25} }}{5} = \frac{5}{5} = 1 \\ \Rightarrow \theta = {45^ \circ } \\ \end{gathered}