 Hello and welcome to the session. Let's work out the following problem. It says show that the lines x minus 5 upon 4 is equal to y minus 7 upon 4 is equal to z plus 3 upon minus 5 and the line x minus 8 upon 7 is equal to y minus 4 upon 1 is equal to z minus 5 upon 3 Intercept each other. So let's now move on to the solution now any point on the line x minus 5 upon 4 is equal to y minus 7 upon 4 is equal to z plus 3 upon minus 5 equal to say k1 so this implies x is equal to 4k1 plus 5 y is equal to 4k1 plus 7 z is equal to 4k1 minus 5k1 minus 3 so any point on the line this is 4k1 plus 5 4k1 plus 7 minus 5k1 minus 3 Now we find any point on the line x minus 8 upon 7 y minus 4 upon 1 is equal to z minus 5 upon 3 Let's put it equal to k2. So from this we have x is equal to 7k2 plus 8 y is equal to k2 plus 4 z is equal to 3k2 plus 5 so any point on the line 2 is 7k2 plus 8 k2 plus 4 3k2 plus 5 now since the two lines intersect therefore the point p is equal to the point q therefore 4k1 plus 5 4 k1 plus 7 minus 5k1 minus 3 is equal to q 7k2 plus 8 k2 plus 4 3k2 plus 5 so this implies 4k1 plus 5 is equal to 7k2 plus 8 or 4k1 minus 7k2 is equal to 5 minus 8 that is 3 and also 4 k1 plus 7 is equal to k2 plus 4 or 4k1 minus k2 is equal to 4 minus 7 that is minus 3 and also minus 5k1 minus 3 is equal to 3k2 plus 5 or we can write it as minus 5k1 minus 3k2 is equal to 5 plus 3 that is 8 let's name this as a this as b and this as c now subtracting equation b from a we have Equation a is 4k1 minus 7k2 is equal to 3 Equation b is 4k1 minus k2 is Equal to minus 3 now since we have to subtract sign will change So plus 4k1 gets cancelled with minus 4k1 minus 7k2 plus k2 is minus 6k2 3 plus 3 is 6 so this implies k2 is equal to minus 1 k2 is minus 1 so now What k2 is equal to minus 1 in any one of the two equations? Let's say in B. So we have 4k1 minus of minus 1 is plus 1 is equal to minus 3 so 4k1 is Equal to minus 3 minus 1 so this implies 4k1 is equal to minus 4 so this implies k1 is equal to minus 1 the value of k1 and k2 is minus 1 so now put k1 is equal to minus 1 and k2 is equal to minus 1 in see question c is minus 5k1 minus 3k2 is equal to 8 So we have minus 5k1 which is minus 1 minus 3 K2 which is minus 1 is equal to 8 So this is 5 plus 3 which is equal to 8 that is 8 is equal to 8 Which is true? Similarly, you can put the values of k1 and k2 in equation a and b and you verify that Ls is will be equal to rhs and that gives us the fact that Any point on the two lines are same that means they intersect each other and that is what we have to prove so this implies the given lines Intersect each other. So this completes the question and the session bye for now. Take care. Have a good day