 Hello and welcome to the session. In this session we will discuss a question which says that final equation of the line which passes through the point of intersection of the lines 2x-y plus 5 is equal to 0 and 5x plus 3y minus 4 is equal to 0 and is perpendicular to the line x minus 3y plus 21 is equal to 0. Now before starting the solution of this question we should know some results. First is for the lines a1x plus b1y plus c1 is equal to 0 and a2x plus b2y plus c2 is equal to 0 the equation of the line through the intersection of the given lines that are these lines is a1x plus b1y plus c1 plus k times a2x plus b2y plus c2 the whole is equal to 0. y is equal to n1x plus c and y is equal to n2x plus c are perpendicular to each other then to n2 is equal to minus 1 where n there is a slope of this line and n2 is the slope of this line that means if these two lines are perpendicular to each other then product of these slopes is equal to minus 1. Now these results will work out as a key idea for solving out this question and now we will start with the solution. Now we have to find the equation of the line through the intersection of these two lines. So given the equations of lines minus y plus 5 is equal to 0 and 5x plus 3y minus 4 is equal to 0. Now let us know this as 1 and this as 2. So of the given lines we will use this result. Now the equation of the line through the intersection of lines 1 and 2 minus y plus 5 is 3y minus whole the whole is equal to minus y plus 5 plus 5kx plus 3ky minus 4k is equal to 0. 5k the whole into x plus 3k minus 1 the whole into y plus 5 minus 4k is equal to 0. Now let us find this as, now let this slope of the line given by equation of the 3 minus coefficient of x which is 2 plus 5k over coefficient of y which is 3k minus equation of the third line is given to us. Also given the equation of the line minus 3y plus 21 is equal to 0. Now let us name this as 4. Now let the slope of the line which is given by equation number 4 is equal to minus coefficient of x which is 1 over coefficient of y which is minus 3. Which is equal to 1 by 3. Now we have to find the equation of the line through the intersection of these two lines and perpendicular also we know that if two lines are perpendicular to each other then product of these slopes is equal to minus 1. This means that the equation of the line through the intersection of the given lines which is the equation number 3 is perpendicular which is given by equation number 4. If the lines given by equation number 3 and 4 perpendicular and this slope of the line 3 is given by m1 and this slope of line 4 is given by we have m1 into m2 is equal to minus 1. So putting the values of m1 and m2 here this implies minus this 5k whole upon 3 to the minus 1 into 1 by 3 is equal to minus 1. Which further implies whole upon 9 to the minus 3 is equal to 1. This implies 2 plus 5k is equal to 9k minus 3 is equal to 3 plus 2. This implies 4 whole is equal to further gives k is equal to 5 by 4. Now let us name the equation of the line through the intersection of line 1 and line 2 its minus y plus 5 5 by 4 into 3 y minus 4 the whole is equal to 0. Multiplying through l by 4 this implies 8 times minus 4 y plus 20 and here it will be 25 x plus 15 y minus 20 is equal to 0 plus 11 y is equal in common this implies 11 into 3 x plus y the whole is equal to 0 which further implies 3 x plus y is equal to 0 by equation of the line equation of this question and that is all for this session hope you all have enjoyed the session.