 Hello and welcome to the session. In this session we will discuss the final question which says, the harmonic mean of two numbers is 4. The activated mean that is A and geometric mean given by G satisfies the relation 2A plus G squared equal to 27 find the two numbers. We consider the two quantities A and B then we have the arithmetic mean that is A and is given by A plus B upon 2 in the same way that is Gm is given by AB and the harmonic mean given by Hm is given by upon A plus B. The mean between the two quantities A and B as 2AB upon A plus B we have the three quantities HP that is 1 upon A, 1 upon H, 1 upon C, AP this means that this would be B upon C. So this means that 1 upon H is equal to 1 upon A plus 1 upon C and this whole upon 2. This means H is equal to B upon 2AC. So we now have H is equal to 2AC upon that is, the harmonic mean between the two numbers A and C is given by 2AC upon A plus C. Similarly, harmonic mean between two numbers A and B is given by 2AB upon A plus B that we use for this question. Before we go on to the solution, we have given the harmonic mean of two numbers and that is 4 in relation that is 2A plus G squared equal to 27. That is A is the arithmetic mean, G is the geometric mean and we are supposed to find the two numbers. We suppose let unit mean upon X plus that is using is 4. So we have 4 is equal to 2XY upon X plus that means the mean is equal to, now 2 times is 4. So where we have 2 into X plus Y the mean is equal to XY. Where does the result? 1. Then next we have two numbers X and Y would be equal to that is A is equal to X plus Y upon 2 which means is equal to, like this the result the geometric mean given by G between the two numbers X and Y is given by square root of XY. So this is equal to square root of XY, G is equal to square root of XY that G squared is equal to XY. Like this the result in relation G squared is equal to 27. This means X plus Y, so X plus Y plus the G squared is XY, so X plus Y plus XY is equal to 27. This is using the results you can see in this result 1 we have XY as 2 into X plus Y the whole inverse in place of XY. We can put 2 into X plus Y the whole minus we have X plus Y plus 2 into X plus Y the whole is equal to 27. And this is using the result 1. This gives us 3 into X plus Y the whole is equal to 27 which means that X plus Y is equal to 27 upon 3. And 3 9 times is 47, thus X plus Y equal to 9. Like this the result using this result 4 minus X plus Y equal to 9 in result 1 2 into X plus Y the whole is equal to XY. We get 2 9 which is 18 that is XY is equal to 18. So where we get Y is equal to 18 upon this result using this result 5 that is Y is equal to 18 is equal to 9 plus 18 is equal to 0. Deciding this quadratic equation we get X squared minus 6X minus 3X plus 18 is equal to 0. For the factorizing we have X into X minus 60 whole minus 3 into X minus 60 whole is equal to 0. The whole into X minus 3 is equal to 0 which means we have X equal to 6 or X equal to 3. So in this result 5 we have Y is equal to 18 upon X is equal to 6 we get Y equal to 18 upon 6. X equal to 6 and Y equal to 3 as one set of values for X and Y is equal to 3 is equal to 18 upon 3 is 18. Therefore X equal to 3 Y equal to 6 is another set of values for X and Y understood the solution of this question.