1. The problem statement, all variables and given/known data This related to coordinate geometry. According to my book it says if there are two intersecting lines L1(a1x+b1y+c1=0) and L2(a2x+b2y+c2=0) Then equation of any line passing through their point of intersection is L1 +p L2 =0 or (a1x+b1y+c1=0) +p(a2x+b2y+c2=0)=0 where p is some constant my problem is that i don't understand what is the proof of this formula.Book also says that (a1x+b1y+c1=0) +p(a2x+b2y+c2=0)=0 represents a Family of lines , and i don't understand what it means too. 2. Relevant equations 3. The attempt at a solution i tried to use some arbitrary line equations and tried to simultaneously solve the equations and get its point of intersection and find the line's equation but i can't relate these 2 methods If k is such that it a1-p.a2=0 how does this proove these lines are concurrent and it equation is (a1x+b1y+c1=0) +p(a2x+b2y+c2=0)=0 book gives the proof as since this line satisfies some P(a,b) hence when we put it into the equation it will yield 0 But that comes only after i have proved this is the equation of all lines passing through common point ,how can this be a proof? Could you please give me a proof of how this formula is derived? Thank you.