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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)
 

Notes:
  • 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
    73698.png
    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
    73686.png




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