 Hello and welcome to the session. In this session we discussed the following question which says find the point on the line x plus 2 upon 3 equal to y plus 1 upon 2 equal to z minus 3 upon 2 and a distance 3 root 2 from the point 1, 2, 3. Before proceeding on to the solution let's see how we can find distance between two points. Consider a point A with coordinates x1, y1, z1 and a point B with coordinates x2, y2, z2 then the distance between the points A and B given by AB is equal to square root of x1 minus x2 the whole square plus y1 minus y2 the whole square plus z1 minus z2 the whole square. This is the key idea for this question. Now let's move on to the solution. The given line is x plus 2 upon 3 equal to y plus 1 upon 2 equal to z minus 3 upon 2. Let this be equal to cylinder. Let this equation be equation 1. Next we have coordinates of any point on line 1 is given by substituting each of these expressions equal to lambda that is we have x plus 2 upon 3 equal to lambda this would give us x equal to 3 lambda minus 2 y plus 1 upon 2 equal to lambda gives us y equal to 2 lambda minus 1 then z minus 3 upon 2 equal to lambda gives z equal to 2 lambda plus 3. We take this point A with coordinates 3 lambda minus 2 2 lambda minus 1 and 2 lambda plus 3 as a point on the line 1. Now the other point given to us is 1 2 3 let this point be point B. Now in the question it's given to us that the distance between the points A and B that is AB is 3 root 2. Now using the key idea we will find the distance between the points A and B. Now we have the coordinates for the point A and also the coordinates for the point B so we have AB is equal to square root of 3 lambda minus 2 minus 1 the whole square plus 2 lambda minus 1 minus 2 the whole square plus 2 lambda plus 3 minus 3 the whole square and this would be equal to 3 root 2. So, from here we get square root of 3 lambda minus 3 the whole square plus 2 lambda minus 3 the whole square plus 2 lambda the whole square equal to 3 root 2. Now, squaring both the sides give us 3 lambda minus 3 the whole square plus 2 lambda minus 3 the whole square plus 2 lambda the whole square equal to 18. This further implies 9 lambda square minus 18 lambda plus 9 plus 4 lambda square minus 12 lambda plus 9 plus 4 lambda square is equal to 18. This further gives us 17 lambda square minus 30 lambda plus 18 equal to 18 that is we get 17 lambda square minus 30 lambda is equal to 0. So, from here we have lambda into 17 lambda minus 30 is equal to 0 that is we have either lambda equal to 0 or 17 lambda minus 30 equal to 0. Thus lambda equal to 0 or lambda equal to 30 upon 17. Now, we have a point A which coordinates 3 lambda minus 2 2 lambda minus 1 2 lambda plus 3. Now, we substitute lambda equal to 0 and also lambda equal to 30 upon 17 1 by 1 in the coordinates of this point. So, for lambda equal to 0 we get the coordinates of the point A as 3 into 0 minus 2 2 into 0 minus 1 2 into 0 plus 3 thus we get the coordinates of the point A as minus 2 minus 1 3. Now, we take lambda equal to 30 upon 17 in the coordinates of the point A that is here. So, we get a point say A dash which coordinates 3 into 30 upon 17 minus 2 2 into 30 upon 17 minus 1 2 into 30 upon 17 plus 3 that is the coordinates of the point A dash are given by 90 upon 17 minus 2 60 upon 17 minus 1 60 upon 17 plus 3 thus we get the coordinates of the point A dash as 56 upon 17 43 upon 17 111 upon 17. So, we say that either point A or the point A dash would lie on the given line at the distance 3 root 2 from the point 1 2 3. So, we have either the point A which coordinates minus 2 minus 1 3 or the point A dash with coordinates 56 upon 17 43 upon 17 and 111 upon 17 lies on the given line at a distance 3 root 2 from the point 1 2 3. So, this completes the session. Hope you have understood the solution for this question.