 Hello friends, let's discuss the following question, it says the slope of a line is double of the slope of another line. If the tangent of angle between them is 1 by 3, find the slopes of the lines. To solve this question we need to know that angle between two lines given by tan theta which is equal to m1 minus m2 upon 1 plus m1 into m2, where m1 and m2 are the slopes of the lines. So this knowledge will work as k idea. Now move on to the solution, we are given that the slope of the one line is double the slope of the other line. So let be the slope of one line, then the slope of another line will be 2m and we are given that the tangent of angle between them is 1 by 3 that is tan theta is equal to 1 by 3. Now we know that tan theta is equal to mod of m1 minus m2 upon 1 plus m1 into m2. Now here m1 is m and m2 is 2m. So we have 1 by 3 is equal to mod of m minus 2m upon 1 plus m into 2m. So this implies 1 upon 3 is equal to mod of minus m upon 1 plus 2m square and this implies 1 by 3 is equal to plus minus minus m upon 1 plus 2m square. Now we have two cases when 1 by 3 is equal to plus minus m upon 1 plus 2m square and when 1 by 3 is equal to minus of minus m upon 1 plus 2m square. Now this implies 1 by 3 is equal to minus m upon 1 plus 2m square and this implies 1 plus 2m square is equal to minus 3m. By cross multiplying we got this and this implies 2m square plus 3m plus 1 is equal to 0 and this implies 2m square plus 2m plus m plus 1 is equal to 0. We just factorize the quadratic equation. Now taking 2m common from the first two terms we have 2m into m plus 1 and taking one common from the last two terms we have 1 into m plus 1 is equal to 0 and this implies m plus 1 into 2m plus 1 is equal to 0. Now this implies m plus 1 is equal to 0 or 2m plus 1 is equal to 0 and this implies m is equal to minus 1 or m is equal to minus 1 by 2. Now we solve this for m. Now this implies 1 by 3 is equal to m upon 1 plus 2m square since minus minus is plus so this is m upon 1 plus 2m square and this implies 1 plus 2m square is equal to 3m and this implies 2m square minus 3m plus 1 is equal to 0. We now factorize this quadratic equation and we solve this for m. So this becomes 2m square minus 2m minus m plus 1 is equal to 0 and this implies taking 2m common from the first two terms we have 2m into m minus 1 and taking minus 1 common from the last two terms we have minus 1 into m minus 1 is equal to 0 and this implies m minus 1 into 2m minus 1 is equal to 0 and this implies m is equal to 1 or m is equal to 1 by 2. So we have four possible values of m and we know that if the slope of one line is m then the slope of the another line is 2m. So for different values of m we find the slope of the line. So if slope of one line m is equal to 1 then the slope of another line is 2m that is 2 since here m is 1. Now if m is 1 by 2 then slope of another line is 2m that is 2 into 1 by 2 that is 1 and if m is minus 1 then the slope of another line is 2 into minus 1 that is minus 2 and if m is equal to minus 1 by 2 then the slope of another line will be 2 into minus 1 by 2 that is minus 1. Hence the slopes of the lines can be 1 and 2 or 1 by 2 and 1 or it can be minus 1 minus 2 or it can be minus 1 by 2 minus 1. So this completes the question. Bye for now. Take care. Have a good day.