 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says the ratio of AM and GM of two positive numbers A and B is M is to N. Show that A is to B is equal to M plus root over M square minus N square is to M minus root over M square minus N square. Let's now start with the solution and there we are given two numbers A and B such that ratio of AM and GM is equal to M is to N. So, arithmetic mean of A and B is A plus B upon 2 divided by and geometric mean of A and B is root over AB. So, this is equal to M upon N which further implies that A plus B upon twice root over AB is equal to N upon N. Now, applying component to endividend and both the sides A plus B plus 2 times root over AB upon A plus B minus 2 times root over AB which is equal to M plus N upon M minus N. Or A can be written as root over A whole square plus root over B whole square plus 2 times root over A into root over B. And in the denominator we can write it as root over A whole square plus root over B whole square minus 2 times root over A into root over B is equal to M plus N upon M minus N which further implies that root over A plus root over B whole square upon root over A minus root over B whole square is equal to M plus N upon M minus N. Taking square root on both the sides we have root over A plus root over B upon root over A minus root over B is equal to root over M plus N upon root over M minus N. Now again taking component to endividend to further have root over A plus root over B plus root over A minus root over B upon root over A plus root over B minus root over A plus root over B is equal to root over M plus N plus root over M minus N upon root over M plus N minus root over M minus N which further implies since minus root B cancels out to plus root B we have 2 times root over A upon 2 times root over B since plus root N minus root A cancels out and here we have root over M plus N plus root over M minus N upon root over M plus N minus root over M minus N 2 cancels out with 2 Now squaring both sides left inside here A upon B and here we have M plus N plus M minus N plus 2 times root over M plus N into root over M minus N and the denominator we have M plus N plus M minus N minus 2 times root over M plus N into root over M minus N. Or this further implies A upon B is equal to N cancels out with minus N here also so in the numerator we have 2M plus 2 times root over M square minus N square upon 2M minus 2 times root over M square minus N square. Or this further implies that A upon B is equal to M plus root over M square minus N square upon M minus root over M square minus N square. Therefore A is to B is equal to M plus root over M square minus N square is to M minus root over M square minus N square. So this completes the session. Take care and have a good day.