General Equation of Pair of Straight Lines

The most general form of a quadratic equation in x and y is

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 ...(4)

Since it is an equation in x and y therefore it must represent the equation of a locus in a plane. It may represent a pair of straight lines if abc + 2fgh â€“ af2 â€“ bg2 â€“ ch2 = 0.

In particular equation of pair of straight lines passing through the origin is

ax2 + 2hxy + by2 = 0 ...(5)

Notes:
• The angle Î¸ between the pair of straight lines is Î¸ =
• The two lines represented by (4) will be parallel if h2 = ab and perpendicular if a b = 0
• The point of intersection of the two lines represented by (4) is
• Bisectors: The equations of the bisectors of the angles between the lines represented by (4) are given by
where (Î±Î²) is the point of intersection of the lines represented by (4).
• Bisectors of angle between the line represented by ax2 + 2hxy + by2 = 0 is
• Distance between the parallel lines: If the two lines represented by (4) are parallel, then the distance between the two parallel lines is given by