I know many such questions have already been asked but this one seemed to be a very messy and a lengthy question for me to solve. The question is as follows
Consider seven different points $P_1,P_2,P_3,P_4,Q_1,Q_2,Q_3$ in plane such that $P_1,P_2,P_3,P_4$ are on straight line $'l'$; and $Q_1,Q_2,Q_3$ are non collinear points and none of them lies on straight line $'l'$( $Q_1,Q_2,Q_3$ are on same side of line $'l'$)
From each of the three points $Q_1,Q_2,Q_3$ perpendicular lines are drawn to the straight lines formed by joining any two of the given points (excluding the point from which the perpendicular line is drawn). Find the maximum possible number of points of intersection of perpendicular lines ( excluding the points $Q_1,Q_2,Q_3$ ).
After a lot of messy work and a turmoil of 3 hours I am getting the answer as $283$. I pretty much think that the answer might be correct. But I wanted to know thoughts of members on this site over how to approach this problem without expending so much of time because such question was asked in our examination papers a few years back and I cannot afford so much time on a single question in exam. So please share your thoughts over this question