 Hi and welcome to the session. Let us discuss the following question. The question says the sum of two numbers is six times their geometric means. Show that numbers are in the ratio 3 plus 2 root 2 is 2 3 minus 2 root 2. Let's now begin with the solution. Let the numbers be a and b according to condition given in the question two numbers is six times their geometric means. The sum of two numbers is equal to six into geometric mean. We have assumed the two numbers as a and b so sum of these two numbers is a plus b and we know the geometric mean between two positive numbers a and b is square root of a b so a plus b is equal to six into square root of a b. Now this implies a plus b upon two into square root of a b is equal to three and three can be written as three by one. Now applying component o and dividend o we get a plus b plus two root a b upon a plus b minus two root a b is equal to three plus one upon three minus one. Now this implies a plus b plus two root a b upon a plus b minus two root a b is equal to four by two. Now this implies square root of a plus square root of b whole square upon square root of a minus square root of b whole square is equal to square root of four by square root of two whole square. Now this implies square root of a plus square root of b upon square root of a minus square root of b is equal to two by root two. We know that two can be written as root two into root two so two by root two is equal to root two into root two by root two and this is equal to root two. So this is equal to root two. Now this implies square root of a plus square root of b upon square root of a minus square root of b is equal to now root two can be written as root two by one. Now again by applying compound angle and dividend o we get a plus square root of b plus square root of a minus square root of b upon square root of a plus square root of b minus square root of a minus square root of b is equal to square root of two plus one upon square root of two minus one. Now this is equal to root a upon two root b so two root a upon two root b is equal to root two plus one upon root two minus one. Now squaring both sides we get a by b is equal to root two plus one whole square upon root two minus one whole square. Now this implies a by b is equal to two plus one plus two root two upon two plus one minus two root two and this implies a by b is equal to three plus two root two upon three minus two root two. Hence we have proved that the numbers are in the ratio three plus two root two is to three minus two root two. This completes the session. Bye and take care.