What is the reason behind the concept of family of lines? Why does putting the coordinates of a random point give the equation of line which is needed? This random point is completely different from the original two lines given to us still it gives the value of the parameter? Why so?

Hi
if first line equation is L₁=0
&second line equation is L₂=0
then L₁+L₂=0 gives us the family of lines or we can say equations of lines which are passing from the point of intersections of given lines (for different values of lambda.
random point will give us lamda value.
Alternatively
we can solve the given equstions of line and get thepoint of intersections
than using two point formula we can find the required equation of line

Thanks and Regards.