 Hello, and welcome to the session I am Deepika here. Let's discuss a question which says, if the times x-1 over minus 3 is equal to y-2 over 2k is equal to z-3 over 2 and x-1 over 3k is equal to y-1 over 1 is equal to z-6 over minus 5 or perpendicular find the value of k. We know that the angle between two lines is given by cos theta is equal to mod of a1 a2 plus b1 b2 plus c1 c2 over under root of a1 square plus b1 square plus c1 square into under root of a2 square plus b2 square plus c2 square where a1 b1 c1 and a2 b2 c2 are the direction ratios two lines and theta is the acute angle between the two lines. So this is a key idea behind that question. We will take the help of this key idea to solve the other question. So let's start the solution. Now given lines are x-1 over minus 3 equal to y-2 over 2k is equal to z-3 over 2 and x-1 over 3k is equal to y-1 over 1 is equal to z-6 over minus 5. So the direction ratios of the first line are minus 3, 2k, 2 and the direction ratios of the second line are 3k, 1 minus 5 so if theta is the angle between two lines then cos theta is equal to mod of minus 3 into 3k plus 2k into 1 plus 2 into minus 5 over under root of minus 3 square plus 2k square plus 2 square into 3k square plus 1 square plus minus 5 square. Now if the lines are perpendicular then theta is equal to 90 degree. Now we know that cos 90 degree is equal to 0 so this implies minus 3 into 3k that is minus 9k plus 2k into 1 that is 2k plus 2 into minus 5 that is minus 10 is equal to 0. So this implies minus 7k minus 10 is equal to 0 again this implies minus 7k is equal to 10 or k is equal to minus 10 over 7 so if the given lines are perpendicular then k is equal to minus 10 over 7. So this is the answer for top of question so this completes our session I hope the solution is clear to you bye and have a nice day.