 Hello and welcome to the session. The question says given that P 3 2 minus 4 and Q 5 4 minus 6 and R 9 8 minus 10 are collinear, find the ratio in which Q divides P R. First it is known the section formula with the help of which we shall be solving the above problem. It says if P with coordinates x1, y1 and z1 and Q with coordinates x2, y2 and z2 are two points R is a point which divides P Q in the ratio m is to n then R with coordinates x, y and z is given by m into x2 plus n into x1 upon m plus n. This is the case when the point R divides the line segment mn internally, m into y2 plus n into y1 upon m plus n and z is m into z2 plus n into z1 upon m plus n. So this is the case when R is a point which divides P Q in the ratio m is to n internally. So with the help of this formula we shall be solving the above problem so this is our key idea. Let us now start with the solution. Here we are given 3 points P Q and R are collinear and Q divides P R. We have to find the ratio in which Q divides P R. Let the point Q divides the line segment P R in the ratio k is to 1. So the coordinates of point Q which are 5, 4 and minus 6 with the help of friction formula are given by 1 into 3 plus k into 9 upon k plus 1 then the y coordinate is given by k into 8 plus 1 into 2 upon k plus 1 and the z coordinate is given by minus 10 into k plus 1 into minus 4 upon k plus 1. And for the simplifying we have 5, 4 minus 6 is equal to 3 plus 9 k upon k plus 1, 8 k plus 2 upon k plus 1 and minus 10 k minus 4 upon k plus 1. Now when comparing the coordinates we have 5 is equal to 3 plus 9 k upon k plus 1, 4 is equal to 8 k plus 2 upon k plus 1 minus 6 is equal to minus 10 k minus 4 upon k plus 1. So with the help of any of these three equations let us find the value of k. With the help of first one let us simplify it we have 5 into k plus 1 is equal to 3 plus 9 k on cos multiplying which is further equal to 5 k plus 5 is equal to 3 plus 9 k or 9 k minus 5 k is equal to 5 minus 3 or we have 4 k is equal to 2 or k is equal to 1 upon 2. Now we have considered the ratio as k is to 1 that is half is to 1 is the ratio. Multiplying both sides of the ratio by 2 we have 1 is to 2 and hence the answer is the required ratio is 1 is to 2 internally. So the point q divides the line p r in the ratio 1 is to 2 internally so this completes the session. Hope you have understood it. Take care and have a good day.