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

In previous tutorials, we saw that the equation of the form 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 between the lines represented by the homogeneous second degree equation as 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 and using the quadratic formula to solve this equation, we have

Let and

Making these substitutions, equations (iii) are and which is the pair of lines represented by the given homogeneous second degree equation.

Now

Also

Since is the angle between the lines, then

Using the values of and 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 . So putting this value in the above equation, we have

**(ii)** If the lines are perpendicular, then . So putting this value in the above equation, we have

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