We studied in previous tutorials that the equation of the form 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 between the lines represented by the homogeneous second degree equation as 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 and using quadratic formula for solving 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, so
Using the values of and 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 , so putting this values in the above equation, we have
(ii) If the line are perpendicular, then , so putting this values in the above equation, we have
This is the condition for two lines to perpendicular to each other.