Lines parallel and perpendicular to a given line:
1. The equation of line parallel to a given line ax + by + c = 0 is ax + by + λ = 0 where λ is constant.
2. 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:
1. 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 \(y-{{y}_{1}}=\left( \frac{m\pm \tan \alpha }{1\mp m\tan \alpha } \right)\left( x-{{x}_{1}} \right)\).
2. 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.
\(\frac{m-{{m}_{1}}}{1+m{{m}_{1}}}=\frac{{{m}_{2}}-m}{1+m{{m}_{2}}}\)
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.