Family of Lines in 2D

3 or more Lines are said to form a family if either

  1. The lines are concurrent i.e. they pass through a common point of intersection
  2. The lines are parallel to each other
Given any 2 Lines in 2D as the following

\(A_1x + B_1y + C_1=0\)
\(A_2x + B_2y + C_2=0\)

The equation of any line that is either parallel to these 2 lines or is concurrent to these 2 lines is given as

\(A_1x + B_1y + C_1 + k(A_2x + B_2y + C_2) = 0\)

The value of the variable \(k\) can be found out when some additional input is given. Following are some additional inputs that are generally given
  1. The point through which the resulting line passes
  2. The line/axis to which the resulting line is parallel or perpendicular