 Hi and welcome to the session I am Shashi and I am going to help you with the following question. Question says in each of the following find the value of k for which the points are collinear. First part is given three points are 7 minus 2, 5, 1, 3, k. First of all let us understand that if the area of a triangle is 0 square units then its vertices are collinear. This is the key idea to solve the given question. Now let us start with the solution we are given three points 7 minus 2, 5, 1 and 3 k. We have to find the value of k so that given three points are collinear. By key idea we know that if three points are collinear then the area of the triangle formed by them is equal to 0. Now we know area of a triangle formed by vertices a x1, y1, b, x2, y2 and c, x3, y3 is given by half multiplied by x1 multiplied by y2 minus y3 plus x2 multiplied by y3 minus y1 plus x3 multiplied by y1 minus y2. Now let us assume that these three points represent points x1, y1, x2, y2, x3, y3. Now clearly we can see value of x1 is equal to 7, value of y1 is equal to minus 2, value of x2 is equal to 5, value of y2 is equal to 1, value of x3 is equal to 3, value of y3 is equal to k. Now we can write area of triangle formed by given points is equal to half multiplied by x1, we know x1 is equal to 7, 7 multiplied by y2 minus y3. Now we know y2 is equal to 1 and y3 is equal to k. So we can write 1 minus k plus x2, you know x2 is equal to 5. So we can write 5 here, multiplied by y3 minus y1, y3 is equal to k and y1 is equal to minus 2. We can write k minus minus 2 here plus x3, you know x3 is equal to 3, multiplied by y1 minus y2, y1 is equal to minus 2 and y2 is equal to 1. So we can write minus 2, minus 1 here. Now this is further equal to 1 upon 2 multiplied by 7 minus 7k plus 5 multiplied by k plus 2 plus 3 multiplied by minus 3. Now simplifying further we get 1 upon 2 multiplied by 7 minus 7k plus 5k plus 10 minus 9. Now this is equal to half multiplied by 8 minus 2k. Now simplifying further we get 4 minus k. So we get area of triangle formed by given points is equal to 4 minus k. Now we know given points are collinear. So the area of the triangle formed by them must be 0 that is 4 minus k is equal to 0. Now adding k on both the sides we get 4 is equal to k or we can simply write it as k is equal to 4. So this is our required answer this completes the session hope you understood the session take care and have a nice day.