# Angle Between the Lines Represented by the Homogeneous Second Degree Equation

In previous tutorials, we saw that the equation of the form $a{x^2} + 2hxy + b{y^2} = 0$ is called the second degree homogeneous equation. And we know that the second degree homogeneous equation represents the pair of straight lines passing through the origin.

Now the angle $\theta$ between the lines represented by the homogeneous second degree equation as $a{x^2} + 2hxy + b{y^2} = 0$ is given as

The second degree homogeneous equation is given as

This equation (i) can be rewritten in the form

Considering the above equation (ii) as a quadratic equation in terms of $y$ and using the quadratic formula to solve 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 is the pair of lines represented by the given homogeneous second degree equation.

Now

Also

Since $\theta$ is the angle between the lines, then

Using the values of ${m_1}$ and ${m_2}$ in the above equation, we have

This completes the proof for an angle between the pair of straight lines.

(i) If the lines are parallel, then $\theta = 0$. So putting this value in the above equation, we have

(ii) If the lines are perpendicular, then $\theta = {90^ \circ }$. So putting this value in the above equation, we have

This is the condition for two lines to be perpendicular to each other.