 Hello and welcome to the session. In this session we discussed the following question which says, the line segment joining the points A with coordinates 2 and 1 and B with coordinates 5 and minus 8 is intersected at the points P and Q such that P is nearer to A. If P also lies on the line given by 2x minus y plus k equal to 0, find the value of k. Consider the points A and B. A has coordinates x1, y1 and point B has coordinates x2, y2. Then point P with coordinates xy divides the line segment joining the points A and B internally in the ratio m1 is to m2. Then the coordinates of the point P are given by x equal to m1 x2 plus m2 x1 upon m1 plus m2 and the y coordinate is equal to m1 y2 plus m2 y1 the whole upon m1 plus m2. This is the key idea that we use for this question. Let's proceed with the solution now. We are given a point A with coordinates 2, 1 and a point B with coordinates 5 minus 8 and it's also given in the question that the line segment joining the points A and B is trisected at the points P and Q. So, consider this figure. Here we have the line segment AB is trisected at the points P and Q. So, we have AP is to PV is equal to 1 is to 2. Now the point P also lies on the line 2x minus y plus k equal to 0. Now using the section formula stated in the key idea, let's find out the coordinates of the point P. We suppose the coordinates of the point P are given by x and y. So, using the section formula we get that x is equal to m1 x2 that is 1 into 5 plus m2 x1 that is 2 into 2 whole upon m1 plus m2 which is 1 plus 2. This is equal to 5 plus 4 upon 3 that is equal to 9 upon 3 equal to 3. Now the y coordinate of the point P is given by y equal to m1 y2 that is 1 into y2 which is minus 8 plus m2 y1 that is 2 into 1 whole upon m1 plus m2 that is 1 plus 2. This is equal to minus 8 plus 2 upon 3 which is equal to minus 6 upon 3 that is equal to minus 2. So, we get y equal to minus 2 plus we have a point P which coordinates 3 minus 2. Now the given line is 2x minus y plus k equal to 0 and we have the point P which coordinates 3 minus 2 lies on the line 2x minus y plus k equal to 0. So, as the point P lies on the given line we'll put 3 in place of x in the given equation of the line and minus 2 in place of y in the given equation of the line. So, we have 2 into 3 minus of minus 2 plus k equal to 0 this gives us 6 plus 2 plus k equal to 0 or we get k equal to minus 8. So, k equal to minus 8 is the required value for k. This completes the session. Hope you have understood the solution of this question.