 Hello and welcome to the session. In this session we discussed the following question which says for what values of P are the points 2, 1, P minus 1 and minus 1, 3, collinear. For the given points A with coordinates x1, y1, B with coordinates x2, y2, C with coordinates x3, y3, area of triangle A, B, C formed by the points A, B and C is given by half into x1 into y2 minus y3 plus x2 into y3 minus y1 plus x3 into y1 minus y2. This is the key idea for this question. Now let's move on to the solution for this question. We consider the point A with coordinates 2, 1, B with coordinates p minus 1 and C with coordinates minus 1, 3. Now since it's given that the points A, B and C are collinear, so the area of the triangle formed by them must be 0. First we find out the area of triangle A, B, C. Now this is equal to half into x1 that is 2 into y2 minus y3 that is minus 1 minus 3 plus x2 that is p into y3 minus y1 that is 3 minus 1 plus x3 that is minus 1 into y1 that is 1 minus y2 that is minus 1. This is further equal to half into 2 into minus 4 plus p into 2 plus minus 1 into 2 that is this is equal to half into minus 8 plus 2p minus 2 that is this further is equal to half into 2p minus 10. So we get area of triangle A, B, C is equal to p minus 5. Since we have already discussed that the area of the triangle formed by the points A, B and C must be 0. So we have p minus 5 equal to 0 gives p equal to 5. Now let's do the verification. So now the points are A with coordinates 2, 1, B with coordinates 5 minus 1 and C with coordinates minus 1, 3. So the area of triangle A, B, C is equal to half into x1 into y2 minus y3 plus x2 into y3 minus y1 plus x3 into y1 minus y2 that is this is equal to half into 2 into minus 4 plus 5 into 2 plus minus 1 into 2 that is this is equal to half into minus 8 plus 10 minus 2 which further is equal to half into 10 minus 10. So this comes out to be equal to 0. So we have got area of triangle A, B, C is 0 thus p equal to 5 is correct. Thus the final answer is for p equal to 5 the points 2, 1, p minus 1, minus 1, 3 are collinear. So this completes the session. Hope you have understood the solution for this question.