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

- The lines are concurrent i.e. they pass through a common point of intersection
- The lines are parallel to each other

\(A_1x + B_1y + C_1=0\)

AND

\(A_2x + B_2y + C_2=0\)

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

- The point through which the resulting line passes
- The line/axis to which the resulting line is parallel or perpendicular