# Angle between the Lines Represented by Homogeneous Second Degree Equation

We studied in previous tutorials 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 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 by

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 is the pair of lines represented by the given homogeneous second degree equation.
Now

Also

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

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

This completes the proof for angle between the pair of straight lines.
(i) If the line are parallel, then $\theta = 0$, so putting this values in the above equation, we have

(ii) If the line are perpendicular, then $\theta = {90^ \circ }$, so putting this values in the above equation, we have

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