 Hello friends and how are you all doing today? The question says the line joining the points having coordinates 2, 1 and 5 minus 8 is trisected at the point P and Q. If point P lies on the line 2x minus y plus k is equal to 0 find the value of k. So let's proceed with the solution that is first understand this question. We are given that a line joining two points that is let it be A and B having coordinates 2, 1 and 5 minus 8 is trisected at the points P and Q. So let this be point P and let this be point Q. Let their coordinates be P's be alpha and beta and Q will be L and M, right? So we are given that if point P lies on the line 2x minus y plus k is equal to 0. We need to find the value of k, right? So here it means that A B upon A B is equal to 1 upon 3. That is this is the this is one of the three equal parts, right? Or we can write it as A P is 2, A B is equal to 1 is 2, 3. So this point alpha is equal to 1 into 5 plus 3 into 2 upon 1 plus 3, isn't it? This is by using the formula that M is equal to Mx2 plus N x1 upon M plus N where is equal to 1 and N is equal to 3, right? On solving we have the formula of sorry, we have the value of alpha as 5 plus 6 upon 4 that is equal to 11 by 4, right? Now similarly we can find the value of beta that is equal to 1 into 8 plus 3 into 1 upon 1 plus 3 that is equal to 8 plus 3 upon 4 that is in turn equal to 1 by 4 itself. This is because we can get the value of y by M y2 plus N y1 upon M plus N where M and N are 1 and 3 respectively. So we can see that therefore we have the coordinates that is alpha beta as 11 by 4 comma 11 by 4. Further, we are given that point P that is having coordinates as alpha and beta lies on the line 2x minus y plus k is equal to 0. Right, we have the value of x and y that is the coordinates of P alpha and beta as 11 by 4 and 11 by 4 respectively. So therefore we have 2 into x that is 11 by 4 minus 11 by 4 that is y plus k equal to 0. This further implies that we have 11 by 2 minus 11 by 4 plus k is equal to 0. On solving it, we have 22 minus upon 4 plus k is equal to 0. This implies 11 by 4 plus k is equal to 0 that in turn gives us the value k as minus 11 by 4. So the required answer to the question is minus 11 by 4 that is the value of k. So this completes the session. Hope you understood the question well and enjoyed it too. Have a nice day ahead.