# Second Degree Homogeneous Equation Represents Pair of Lines

As we know that the equation of the form $a{x^2} + 2hxy + b{y^2} = 0$ is called the second degree homogeneous equation.
The second degree homogeneous equation represents the pair of straight lines passing through the origin.
The second degree homogeneous equation is given as

This equation (i) can be rewritten of the form

Considering the above equation (ii) as quadratic equation in terms of $y$ and using quadratic formula for solving this equation, we have

Let ${m_1} = \frac{{ - h + \sqrt {{h^2} - ab} }}{b}$ and ${m_2} = \frac{{ - h - \sqrt {{h^2} - ab} }}{b}$
Making these substitutions, equations (iii) are $y = {m_1}x$ and $y = {m_2}x$ which are obviously equations of lines passing through origin because there is no y-intercepts in these equation (iii).
This shows that the second degree homogeneous equation represents the pair of straight lines passing through the origin.