 Good morning friends, I am Purva and today we will discuss the following question. Find the values of p so that the lines 1-x upon 3 is equal to 7y-14 upon 2p is equal to z-3 upon 2 and 7-7x upon 3p is equal to y-5 upon 1 is equal to 6-z upon 5 are at right angles. Let us begin with the solution now. Now we are given the equation of the lines as 1-x upon 3 is equal to 7y-14 upon 2p is equal to z-3 upon 2 and 7-7x upon 3p is equal to y-5 upon 1 is equal to 6-z upon 5. Or we can rewrite the equations as x-1 upon minus 3 is equal to now dividing the numerator and denominator by 7 we get y-2 upon 2p upon 7 is equal to z-3 upon 2 and again dividing the numerator and denominator by 7 we get x-1 upon minus 3p upon 7 we take out minus common so we get x-1 upon minus 3p upon 7 is equal to y-5 upon 1 is equal to again taking out minus common we get z-6 upon minus 5. Now the direction ratios for the line 1 that is a-1 b-1 and c-1 are minus 3 2p upon 7 and 2 and the direction ratios for line 2 that is a-2 b-2 and c-2 are minus 3p upon 7 1 and minus 5. Now the angle between the two lines is given by cos theta is equal to mod of a-1 a-2 that is minus 3 into minus 3p upon 7 plus b-1 b-2 that is 2p upon 7 into 1 plus c-1 c-2 that is 2 into minus 5 upon under root of a-1 square that is minus 3 square plus b-1 square that is 2p upon 7 whole square plus c-1 square that is 2 square into under root of minus 3p upon 7 square plus b-2 square that is 1 square plus c-2 square that is minus 5 square. Now we are given that the lines are at right angles so for the lines to be at right angles numerator should be equal to 0. So we get minus 3 into minus 3p upon 7 plus 2p upon 7 into 1 plus 2 into minus 5 is equal to 0. And this implies minus 3 into minus 3p is equal to 9p upon 7 plus 2p upon 7 into 1 is equal to 2p upon 7 2 into minus 5 is equal to minus 10 is equal to 0. This implies 11p upon 7 is equal to 10 which further implies 11p is equal to 70 and this implies p is equal to 70 upon 11. So we have got the value of p as 70 upon 11. This is our answer. Hope you have understood the solution. Bye and take care.