The value of A is 1/5 and 10 G2=AH =6×6=36 then in the given equation put the value of H and solved it and got the answer

Given:
The geometric mean of two numbers is 6.

To Find:
The value of A, the arithmetic mean of the two numbers.

Approach:
We are given the geometric mean of two numbers, which means that the square root of their product is 6. Let's assume the two numbers to be x and y.
So, we have the equation:

sqrt(xy) = 6 ---(1)

We are also given that the arithmetic mean A and harmonic mean H of the two numbers satisfy the equation:

90A + 5H = 918 ---(2)

We need to find the value of A.

Solution:

Step 1: Finding the values of x and y

From equation (1), we have:

sqrt(xy) = 6

Squaring both sides, we get:

xy = 36

Now, we have one equation with two variables. We need another equation to solve for x and y.

Step 2: Solving for x and y using equation (2)

From equation (2), we have:

90A + 5H = 918

We know that the arithmetic mean A of two numbers is given by:

A = (x + y) / 2

And the harmonic mean H is given by:

H = 2xy / (x + y)

Substituting these values into equation (2), we get:

90[(x + y) / 2] + 5[2xy / (x + y)] = 918

Simplifying this equation, we get:

45(x + y) + 10xy / (x + y) = 918

Multiplying through by (x + y), we get:

45(x + y)^2 + 10xy = 918(x + y)

Expanding and rearranging terms, we get:

45x^2 + 90xy + 45y^2 + 10xy - 918x - 918y = 0

Simplifying further, we get:

45x^2 + 100xy + 45y^2 - 918x - 918y = 0

Factoring out common terms, we get:

(45x^2 - 918x) + (100xy - 918y) + (45y^2) = 0

Taking out common factors, we get:

45x(x - 34) + 100y(x - 9) + 45y^2 = 0

Now, we have two equations:

45x(x - 34) = 0 ---(3)
100y(x - 9) = 0 ---(4)

Step 3: Solving equations (3) and (4) to find the values of x and y

From equation (3), we have two possibilities:

1. 45x = 0, which implies x = 0
2. x - 34 = 0, which implies x = 34

From equation (4), we have two possibilities:

1. 100y = 0, which implies y = 0
2. x - 9 = 0, which implies x = 9

So, the possible values of x and y are:
1. x