Lines through Points of Intersection of Two Lines

Lines parallel and perpendicular to a given line:

• The equation of line parallel to a given line ax + by + c = 0 is ax + by + λ = 0 where λ is constant.
• The equation of line perpendicular to a given line ax + by + c = 0 is bx – ay + λ = 0 where λ is constant.

Equation of straight lines passing through a given point and making a given angle with a given line:

• The equation of the straight lines which pass through a point (x1, y1) and make a given angle α with straight line y = mx + c is y - y₁ = {[m ± tan α]/ [1 ± m tan α]}(x - x₁).
• Let there be two lines with slopes m1 and m2 meet at point P and a line having slope m bisects the two lines with angle θ. (m - m₁) / (1 + mm₁) = (m₂ - m)/ (1 + mm₂)
  

  

Family of lines through the intersection of two given lines:

• The equation of family of lines passing through the intersection of lines a₁ x + b₁ y + c₁ = 0 and a₂x + b₂ y + c₂= 0 is a₁x + b₁ y + c₁ + λ (a₂x + b₂ y + c₂) = 0 where λ is a parameter.