 Hi, and welcome to the session. Let's work out the following question. The question says the shot fired at a mark in a horizontal plane through the point of projection goes a meter beyond it when the angle of elevation is alpha. When the angle of elevation is beta, it falls b meter short of the mark. Show that the proper elevation to hit the mark is half of sine inverse a sine 2 beta plus a sine 2 alpha divided by alpha plus beta. So now let us see the solution to this question. So let the point a and the point b be the points where the shot fell at distance a and b when projected at angles alpha and beta respectively. Let c is the exact point at a distance r to be hit. Therefore r plus a is equal to u square sine 2 alpha divided by g. We call this equation 1. r minus b is equal to u square sine 2 beta divided by g. We call this 2 and r is equal to u square sine 2 theta divided by g. We call this 3. Now subtract 2 from 1. We get a plus b is equal to u square upon g into sine alpha minus sine beta. We call this equation 4. Sorry, here we see that this is sine 2 alpha and sine 2 beta. Now divide 1 by 2 and we get r plus a divided by r minus b is equal to sine 2 alpha divided by sine 2 beta. This implies r sine 2 beta plus a sine 2 beta is equal to r sine 2 alpha minus b sine 2 alpha. This we get by cross multiplying this. This implies r into sine 2 alpha minus sine 2 beta is equal to a sine 2 beta plus b sine 2 alpha. Therefore r is equal to a sine 2 beta plus b sine 2 alpha divided by sine 2 alpha minus sine 2 beta and we call this 5. Now from equation 3 and equation 5 we get r is equal to u square upon g sine 2 theta that is same as a sine 2 beta plus b sine 2 alpha divided by sine 2 alpha minus sine 2 beta we call this 6. Now from 4 and 6 we get a plus b divided by sine 2 alpha minus sine 2 beta into sine 2 theta is equal to a sine 2 beta plus b sine 2 alpha divided by sine 2 alpha minus sine 2 beta. Now we see that sine 2 alpha minus sine 2 beta gets cancelled with sine 2 alpha minus sine 2 beta on the right hand side. This implies sine 2 theta is equal to a sine 2 beta plus b sine 2 alpha divided by a plus b. This implies theta is equal to half of sine inverse a sine 2 beta plus b sine 2 alpha divided by a plus b. So this is our answer to this question or we can say that this is what we were supposed to prove in this question. So I hope that you understood the solution and enjoyed the session. Have a good day.