 Hello and welcome to the session. Let's discuss the following question. It says forces P and Q at an angle alpha if P is doubled and Q remains the same, the resultant is also doubled. Show that for P sine alpha is equal to the square root of 16 P square minus 9 Q square. So let's now move on to the solution. We are given that the angle between Q and Q is alpha and let R be the resultant of the forces P and Q. Then R square is given by P square plus Q square plus 2 PQ cos alpha. Let us name this as 1. Now we are given that if P is doubled, Q remains the same resultant is doubled. P is doubled, Q remains the same then resultant is doubled. So we have 2R square is equal to 2P square because P is doubled but Q remains the same, 2 into 2P into Q to cos alpha. So we have 4R square is equal to 4P square plus Q square plus 4PQ cos alpha. Let us name this as 2. Now we will substitute the value of R given by 1 into, so put 1 into, we have 4 into R square. R square is P square plus Q square plus 2PQ cos alpha is equal to 4P square plus Q square plus 4PQ cos alpha. So we have 4P square plus 4Q square plus 8PQ cos alpha is equal to 4P square plus Q square plus 4PQ cos alpha. Now 4P square gets cancelled with 4P square and 4Q square minus Q square is 3Q square. 4PQ cos alpha minus 8PQ cos alpha is minus 4PQ cos alpha cancelling. Q on both sides we have 3Q is equal to minus 4P cos alpha. So we have cos alpha is equal to 3Q upon minus 4P. Now sin alpha is given by the square root of 1 minus cos square alpha. So this is equal to the square root of 1 minus cos alpha is minus 3Q by 4P. So it is minus 3Q by 4P square. This is equal to the square root of 1 minus 9Q square upon 16P square. This is again equal to 16P square minus 9Q square upon 16P square. This is again equal to 16, the square root of 16, P square minus 9Q square upon 4P. So this implies 4P sin alpha is equal to the square root of 16, P square minus 9Q square. And this is what we had to prove. So this completes the question and the session. Bye for now. Take care. Have a good day.