 Hello and welcome to the session. In this session we will discuss a question which says that Find the equation of the line which research the obtuse angle between the lines x-2y plus 4 is equal to 0 and 4x-3y plus 2 is equal to 0. Now before starting the solution of this question, we should know some results. And the first result is for the lines a1x plus b1y plus 0 is equal to 0 and a2x plus b2y plus c2 is equal to 0. Let us think this as 1 and this as 2. If theta be the angle between the lines 1 and 2 then tan theta is equal to mod of m1 minus m2 over 1 plus m1 into m2 where m1 is the slope of the line which is given by equation number 1 and m2 is the slope of line given by equation number 2. Secondly the equations by sectors between the lines which is given by equation number 1 and 2 is given by plus b1y plus c1 whole upon square root of a1 square plus v1 square is equal to plus minus a2x plus b2y plus c2 whole upon square root of a2 square plus b2 square. Where the equation obtained on taking the positive sign, the equation of the bisector, that bisector on taking the negative sign, the equation obtained is the equation. These results will work out as the key idea for solving out this question with the solution. Here the equation of the lines are given to us. So given the equations of the lines as 4 is equal to 0 and 4x minus 3y plus 2 is equal to 0. Now let us think which as 1 and this as this result which is given in the key idea, the equations lines which is given by equation number 1 and 2, square root of a1 square plus minus 2 square is equal to plus minus 3y plus 2 whole upon 4 square plus minus 3 square. Here a1 is 1, b1 is minus 2 and c1 is 4 and here a2 is 4, b2 is minus 3 and c2 is 2. This further implies 2y plus 4 whole upon square root of 1 plus 4 is equal to plus minus 4x minus 3y plus 2 whole upon square root of 16 plus 9. Which further implies whole upon root 5 is equal to plus minus 4x minus 3y plus 2 whole upon root 25. Further implies root 5 is equal to plus minus 4x minus 3y plus 2 whole upon, now this implies 2y plus 4 whole upon is equal to plus minus 4x minus 3y plus 2 over, now here root 5 will be cancelled with root 5. So this implies x minus 2y plus 4 is equal to plus minus 4x minus 3y plus 2 whole upon. Now the equation of the bisector of the angle it will be x minus 2y plus is equal to plus minus 3y plus 2 whole upon. Which implies on cross multiplying root 5 into x minus 2y plus 4 whole is equal to 4x minus 3y plus 2. Further implies root 5x root 5y plus 4 minus 4x plus 3y minus 2 is equal to 0. Which further implies minus root 5 the whole into x 3 minus 2 root 5 the whole into y plus equal to 0. Now let us name this equation as equation number 3. Now the other bisector now again in this equation on taking the negative sign it will be x minus 2y plus 4 is equal to minus 3y plus 2 whole upon. On cross multiplying this implies it into x minus 2y plus 4 the whole is equal to minus 4x plus 3y minus 2. Which further implies root 5x minus 2 root 5y plus 4 root 5 plus 4x minus 3y plus 2 is equal to 0. Which further implies 5 the whole into x minus 5 the whole into y plus is equal to 0. And let us name this as equation number 4. Now we have this as equation number 1 and this as equation number 2. Now is equal to minus x which is 1 over coefficient of y which is minus 2 which is equal to 1 by 2. Now this is the equation number 3. Now slope m3 of the line which is given by equation number 3 is equal to minus of x which is 4 minus root 5 whole upon coefficient of y which is minus of 3 minus 2 root 5. Which is equal to 4 minus root 5 whole upon 3 minus 2 root 5. Now this is the equation number 4. Now slope m4 of the line which is given by equation number 4 is equal to minus coefficient of x which is 4 plus root 5 over coefficient of y which is minus of 3 plus 2 root 5 which is further equal to 4 plus root 5 whole upon 3 plus 2 root 5. Now using this result which is given in the key idea if theta be the angle between the lines which is given by equation number 1 and 3. Then tan theta is equal to mod of m1 minus m3 whole upon 1 plus m1 into m3. Now this is the value of m1 is the value of m3. So if we do the values here this will be equal to mod of 1 by 2 minus minus root 5 whole upon 3 whole upon 1 plus 1 by 2 into 4 minus root 5 whole upon. Taking the regime in the numerator and denominator this will be equal to mod of 3 minus 2 root 5 minus 8 plus 2 into 3 minus 2 root 5 the whole whole upon 6 minus 4 root 5 plus 4 minus root 5 whole upon 2 into 3 minus 2 root 5 the whole. Now this will be equal to mod of, now these terms will be cancelled with each other so it will be and if the denominator it will be 10 minus 5 root 5. This will be equal to mod of minus 5 over in the denominator taking minus 5 common given that it will be root 5 minus 2. Now these terms will be cancelled with each other so it will be equal to 1 over root 5 minus 2 root 5 minus 2. Now rationalizing 10 theta is equal to 1 over root 5 minus 2 into root 5 plus 2 over root 5 plus 2. It is equal to root 5 plus 2 over A minus B into A plus B is 8 plus minus B square so it will be 5 minus 4 which is equal to root 5 plus 2 by 1 which is equal to root 5 plus 2. Now common theta is greater than 1 is greater than 1045 degrees which further implies theta is greater than 45 degrees which implies on multiplying both sides by 2 2 theta is greater than 90 degrees. Now we have taken theta as the angle between the lines 1 and 3 is greater than 90 degrees. Therefore the equation 3 that is this equation is the bisector of the obtuse angle. Therefore the equation number 3 that is the hole into x if the hole into y plus if is equal to 0 is the bisector of obtuse angle between the given lines. And that's all for this session. Hope you all have enjoyed this session.