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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 + 2fghaf2bg2ch2 = 0.
In particular equation of pair of straight lines passing through the origin is
ax2 + 2hxy + by2 = 0 ...(5)

  • The angle θ between the pair of straight lines is θ = 73710.png
  • 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 73704.png
  • Bisectors: The equations of the bisectors of the angles between the lines represented by (4) are given by
    where (αβ) is the point of intersection of the lines represented by (4).
  • Bisectors of angle between the line represented by ax2 + 2hxy + by2 = 0 is 73692.png
  • 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

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