If the harmonic mean between two positive numbers is to their G.M. as 12 : 13, the numbers are in the ratio

Options

A. `12 : 13`
B. `(1)/(12) : (1)/(13)`
C. `4 : 9`
D. `(1)/(4) : (1)/(9)`

Answer

To solve the problem of finding the ratio of two positive numbers \( A \) and \( B \) given that their harmonic mean (H) to their geometric mean (G) is in the ratio 12:13, we can follow these steps:

### Step 1: Define the Harmonic Mean and Geometric Mean
The harmonic mean \( H \) of two numbers \( A \) and \( B \) is given by:
\[ H = \frac{2AB}{A + B} \]
The geometric mean \( G \) of \( A \) and \( B \) is given by:
\[ G = \sqrt{AB} \]

### Step 2: Set Up the Ratio
According to the problem, we have:
\[ \frac{H}{G} = \frac{12}{13} \]
Substituting the expressions for \( H \) and \( G \):
\[ \frac{\frac{2AB}{A + B}}{\sqrt{AB}} = \frac{12}{13} \]

### Step 3: Simplify the Equation
This can be rewritten as:
\[ \frac{2AB}{(A + B) \sqrt{AB}} = \frac{12}{13} \]
Cross-multiplying gives:
\[ 2AB \cdot 13 = 12(A + B) \sqrt{AB} \]
This simplifies to:
\[ 26AB = 12(A + B) \sqrt{AB} \]

### Step 4: Rearranging the Equation
Dividing both sides by \( \sqrt{AB} \) (assuming \( AB \neq 0 \)):
\[ 26\sqrt{AB} = 12\left(\frac{A + B}{\sqrt{AB}}\right) \]
Let \( T = \frac{A + B}{\sqrt{AB}} \). Then we have:
\[ 26 = 12T \]
This implies:
\[ T = \frac{26}{12} = \frac{13}{6} \]

### Step 5: Expressing \( T \) in Terms of \( A \) and \( B \)
We know that:
\[ T = \frac{A}{\sqrt{AB}} + \frac{B}{\sqrt{AB}} = \frac{\sqrt{A}}{\sqrt{B}} + \frac{\sqrt{B}}{\sqrt{A}} \]
Let \( x = \frac{\sqrt{A}}{\sqrt{B}} \). Then:
\[ T = x + \frac{1}{x} \]
Setting this equal to \( \frac{13}{6} \):
\[ x + \frac{1}{x} = \frac{13}{6} \]

### Step 6: Multiplying Through by \( 6x \)
Multiplying through by \( 6x \) gives:
\[ 6x^2 - 13x + 6 = 0 \]

### Step 7: Solving the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{13 \pm \sqrt{(-13)^2 - 4 \cdot 6 \cdot 6}}{2 \cdot 6} \]
\[ x = \frac{13 \pm \sqrt{169 - 144}}{12} \]
\[ x = \frac{13 \pm 5}{12} \]
This gives us two solutions:
1. \( x = \frac{18}{12} = \frac{3}{2} \)
2. \( x = \frac{8}{12} = \frac{2}{3} \)

### Step 8: Finding the Ratios
The ratios of \( A \) and \( B \) can be derived from \( x \):
1. If \( x = \frac{3}{2} \), then \( \frac{A}{B} = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \)
2. If \( x = \frac{2}{3} \), then \( \frac{A}{B} = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \)

### Final Answer
Thus, the numbers \( A \) and \( B \) are in the ratio:
\[ \frac{9}{4} \quad \text{or} \quad \frac{4}{9} \]