 Hi, and welcome to the session. Let us discuss the following question. The question says if the lines y is equal to 3x plus 1 and 2y is equal to x plus 3 are equally inclined to the line, y is equal to nx plus 4, find the value of n. In this question, lines p, q, and pr, whose equations are y is equal to 3x plus 1 and 2y is equal to x plus 3 are equally inclined to the line, q, r, whose equation is y is equal to mx plus 4. Now equally inclined means the angle between p, q, and qr, that is p, q, r is equal to angle between pr and qr, that is pr, q. Let's now begin with the solution. We will first find the angle between lines p, q, and qr. Slope of line p, q is 3, right, and slope of line qr is n. So angle between lines p, q, and qr is equal to what of m minus 3 upon 1 plus 3a. Let's name this equation as equation number 1. Now we will find the angle between lines pr and qr. Equation of line pr is 2y is equal to x plus 3. Now 2y is equal to x plus 3 can be written as y is equal to 1 by 2 into x plus 3 by 2. Comparing this equation with y is equal to mx plus c, we find that slope of line pr is 1 by 2. So angle between qr is theta is equal to mod of m minus 1 by 2 upon 1 plus 1 by 2 into m. Let's name this equation as equation number 2. From 1 and 2, we have m minus 3 upon 1 plus 3m is equal to plus minus m minus 1 by 2 upon 1 plus 1 by 2 into m. Now first consider m minus 3 upon 1 plus 3m is equal to m minus 1 by 2 upon 1 plus m by 2. Now this implies m minus 3 upon 1 plus 3m is equal to 2m minus 1 upon 2 plus m. This implies m minus 3 into m plus 2 is equal to 2m minus 1 into 3m plus 1. This implies m squared minus m minus 6 is equal to 6m squared minus m minus 1. This implies minus 5m squared is equal to 5. And this implies m squared is equal to minus 1. This implies m is equal to square root of minus 1. And this cannot be possible because slope of a line cannot be a complex number. Now consider m minus 3 upon 1 plus 3m is equal to minus into 2m minus 1 upon m plus 2. Now this implies m minus 3 into m plus 2 is equal to minus 2m minus 1 into 1 plus 3m. And this implies m squared minus m minus 6 is equal to minus 6m squared plus m plus 1. And this implies 7m squared minus 2m minus 7 is equal to 0. Now we will solve this equation by using this criminate. So m is equal to 2 plus minus square root of 4 minus 4 into 7 into minus 7 upon 40. This is equal to 2 plus minus square root of 4 plus 196 upon 40. And this is equal to 2 plus minus square root of 200 upon 40. This is equal to 2 plus minus 10 root 2 upon 14. And this is equal to 1 plus minus 5 root 2 upon 7. Hence the required value of m is 1 plus minus 5 root 2 upon 7. This is our required answer. So this completes the session. Bye and take care.