 Hello 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. Given points are 8, 1, k minus 4, 2 minus 5. First of all let us understand that if the area of a triangle is 0 square units then its vertices will be collinear. This is the k idea to solve the given question. Now let us start the solution we are given points 8, 1, k minus 4, 2 minus 5 we have to find the value of k so that all the three points are collinear. We know that the 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 the points x1, y1, x2, y2, x3, y3. Now clearly we can see value of x1 is equal to 8, value of y1 is equal to 1, value of x2 is equal to k, value of y2 is equal to minus 4, value of x3 is equal to 2 and value of y3 is equal to minus 5. Now area of a triangle formed by given points is equal to half multiplied by x1, x1 we know is equal to 8, so 8 multiplied by y2 minus y3, y2 is equal to minus 4, minus y3 is equal to minus 5, so we get 8 multiplied by minus 4 minus minus 5 plus x2, we know x2 is equal to k, so we will write k multiplied by y3 minus y1, y3 is equal to minus 5 and y1 is equal to 1 plus we know x3, x3 is equal to 2, 2 multiplied by y1 minus y2, y1 is equal to 1 and y2 is equal to minus 4, so we can write 1 minus minus 4, now this is further equal to 1 upon 2 multiplied by 8 multiplied by minus 4 plus 5 plus k multiplied by minus 6 plus 2 multiplied by 1 plus 4, here these 2 negative signs will change into positive sign, we know minus 5 minus 1 is equal to minus 6 and here again these 2 negative signs will change into positive sign, now this is further equal to 1 upon 2 multiplied by 8 multiplied by 1 we know minus 4 plus 5 is equal to 1 minus 6k plus we know 1 plus 4 is equal to 5 and 5 multiplied by 2 is equal to 10, now we get 1 upon 2 multiplied by 18 minus 6k, now this is further equal to 9 minus 3k, now we get area of triangle formed by given points is equal to 9 minus 3k, now we know the given points are collinear, so the area of the triangle formed by them must be 0, so we can write since the given points are collinear the area of the triangle formed by them must be 0 that is 9 minus 3k is equal to 0, now adding 3k on both the sides we get 9 is equal to 3k, now dividing both the sides by 3 we get 3 is equal to k or we can simply write k is equal to 3, so this is our required answer this completes the session hope you understood the session take care and have a nice day.