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