 Hello friends, let's discuss the following question. It says, find the equation of the lines through the points 3, 2 which make an angle of 45 degrees with the line x-2y is equal to 3. To solve this question we need to know the equation of line passing through a point say x1, y1 and having slope m. The equation is given by y-y1 is equal to m into x-x1. So this knowledge will work as t-idea. Let us now proceed on with the solution. Given line is x-2y is equal to 3. This implies y is equal to 1 by 2 x-3 by 2. So the slope of the given line is 1 by 2. Now we have to find the equation of the line which passes through the point 3, 2 and which makes an angle of 45 degrees with the line x-2y is equal to 3. Now let m be the slope of the required line. It has slope 1 by 2 and the required line has slope m. And we know that tan theta is equal to mod of m1 minus m2 upon 1 plus m1 into m2 where m1 and m2 are the slopes of the given lines. Now we know that the given line and the required line makes an angle of 45 degrees. Therefore, tan 45 degrees is equal to mod of m minus 1 by 2 upon 1 plus m into 1 by 2. And this implies 1 is equal to plus minus m minus 1 by 2 upon 1 plus m upon 2. Now we have 2 cases when 1 is equal to m minus 1 by 2 upon 1 plus m by 2 when 1 is equal to minus of m minus 1 by 2 upon 1 plus m by 2. Let us first solve this. We have 1 is equal to 2 m minus 1 upon 2 plus m taking LCM and this implies 2 plus m is equal to 2 m minus 1 and this implies m is equal to 3. Now we solve this. We have 1 is equal to minus of 2 m minus 1 upon 2 plus m and this implies 2 plus m is equal to minus of 2 m minus 1 and this implies 2 plus m is equal to minus 2 m plus 1 and this again implies 3 m is equal to 1 minus 2 and this implies m is equal to minus 1 by 3. Now we have 2 values of m. So the equation of line passing through the point 3 2 and having slope is given by y minus 2 is equal to 3 into x minus 3. This implies y minus 2 is equal to 3 x minus 9. This implies y minus 3 x is equal to minus 7 and this implies 3 x minus y is equal to 7. Now we have to find the equation of line passing through the point 3 2 and having slope minus 1 by 3 and its equation is given by y minus 2 is equal to minus 1 by 3 into x minus 3 and this implies 3 into y minus 2 is equal to minus x plus 3. We just opened the bracket and cross multiplied and this implies 3 y minus 6 is equal to minus x plus 3 and this implies 3 y plus x is equal to 9. We just transposed minus 6 to the right hand side and minus x to the left hand side and this implies x plus 3 y is equal to 9. Hence the required lines are 3 x minus y is equal to 7 and x plus 3 y is equal to 9. So this completes the question. Hope you enjoyed the session. Goodbye and take care.