If arithmetic mean of two positive numbers is A, their geometric mean is G and harmonic mean H, then H is equal to

Options

A. `(G^(2))/(A)`
B. `(A^(2))/(G^(2))`
C. `(A)/(G^(2))`
D. `(G)/(A^(2))`

Answer

To find the harmonic mean \( H \) of two positive numbers given their arithmetic mean \( A \) and geometric mean \( G \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the two positive numbers**:
Let the two positive numbers be \( x \) and \( y \).

2. **Express the Arithmetic Mean**:
The arithmetic mean \( A \) is given by the formula:
\[ A = \frac{x + y}{2} \]
From this, we can express \( x + y \):
\[ x + y = 2A \]

3. **Express the Geometric Mean**:
The geometric mean \( G \) is given by:
\[ G = \sqrt{xy} \]
Squaring both sides, we get:
\[ G^2 = xy \]

4. **Express the Harmonic Mean**:
The harmonic mean \( H \) is defined as:
\[ H = \frac{2xy}{x + y} \]
Substituting the expressions for \( xy \) and \( x + y \) that we derived earlier:
\[ H = \frac{2xy}{x + y} = \frac{2G^2}{2A} \]

5. **Simplify the Expression**:
The \( 2 \) in the numerator and denominator cancels out:
\[ H = \frac{G^2}{A} \]

### Final Result:
Thus, the harmonic mean \( H \) in terms of the arithmetic mean \( A \) and geometric mean \( G \) is:
\[ H = \frac{G^2}{A} \]