 Hello friends, let's discuss the following question. It says if three lines whose equations are y is equal to m1 x plus c1, y is equal to m2 x plus c2 and y is equal to m3 x plus c3 are concurrent, then show that m1 into c2 minus c3 plus m2 into c3 minus c1 plus m3 into c1 minus c2 is equal to 0. Before moving on to the solution, first understand the basic approach to solve this question. We are given that the three lines are concurrent. That means they pass through one point. That means the three lines intersect at one point. So we will first find the intersection point of any two lines and since the three lines are given to be concurrent, third line will also pass through the point of intersection of the two lines. Keeping this approach in mind, let us now move on to the solution. We are given three lines y is equal to m1 x plus c1 and y is equal to m2 x plus c2. Now we find point of intersection of these two lines. Let's solve this as one and this as two. Now subtract 2 from 1, we have y is equal to m1 x plus c1, y is equal to m2 x plus c2. Subtracting, we get 0 is equal to m1 minus m2 plus c1 minus c2 and this implies x is equal to minus c1 minus c2 upon m1 minus m2. Now we put x is equal to minus c1 minus c2 upon m1 minus m2 in any one of the two equations, equation one or equation two. So let's put it in equation one. So we have y is equal to m1 into x that is minus minus c1 minus c2 upon m1 minus m2 plus c1. So this implies minus m1 into c1 minus c2 plus c1 into m1 minus m2 upon m1 minus m2 is equal to m3 into x which is minus of c1 minus c2 upon m1 minus m2 plus c3 and this implies minus m1 into c1 minus c2 plus c1 into m1 minus m2 is equal to minus m3 into c1 minus c2 plus c3 into m1 minus m2. We just took the LCM and cancelled m minus and this implies y is equal to minus m1 into c1 minus c2 plus c1 into m1 minus m2 upon m1 minus m2 by taking the LCM. Therefore the point of intersection of the line 1 and 2 is minus of c1 minus c2 upon m1 minus m2 and minus m1 into c1 minus c2 plus c1 into m1 minus m2 upon m1 minus m2. Let's call this as A. Now the third equation of the line is given to us as y is equal to m3 x plus c3 and we are given that the three lines are concurrent. Concurrent means they pass through one point that means the third line passes through the point of intersection of the line 1 and 2 that means A satisfies the equation of the third line. So we put x is equal to minus of c1 minus c2 upon m1 minus m2 and y is equal to minus m1 into c1 minus c2 plus c1 into m1 minus m2 upon m1 minus m2 in 3. So we have minus m1 into c1 minus c2 plus c1 into m1 minus m2 upon m1 minus m2 is equal to m3 into x which is minus of c1 minus c2 upon m1 minus m2 plus c3. Now this implies minus m1 into c1 minus c2 plus c1 into m1 minus m2 upon m1 minus m2 is equal to minus m3 into c1 minus c2 plus c3 into m1 minus m2 upon m1 minus m2. Now we can say m1 minus m2 from the both sides in the denominator we have minus m1 into c1 minus c2 plus c1 into m1 minus m2 is equal to minus m3 into c1 minus c2 plus c3 into m1 minus m2. Now this implies minus m1 into c1 minus c2 plus c1 into m1 minus m2 plus m3 into c1 minus c2 minus c3 into m1 minus m2 is equal to 0. Now this can be further written as m1 into c2 minus c1 taking minus in common plus c1 into m1 minus m2 plus m3 into c1 minus c2 plus c3 into m2 minus m1 is equal to 0 and this again implies m1 into c2 minus c1 plus c1 minus c3 we are just collecting the terms with m1 now we collect the terms with m2 with m2 into minus c1 plus c3 now we collect the terms with m3 so m3 into c1 minus c2 and this is equal to 0. Now this is again equal to m1 into c2 minus c3 plus m2 into c3 minus c1 plus m3 into c1 minus c2 is equal to 0 and this is what we have to prove hence the result is proved. So this completes the question I hope you enjoyed this session goodbye and take care.