If the A.M between a and b is m times their H.M then a:b is

Options

A. `(sqrtm + sqrt(m+1))/(sqrtm-sqrt(m+1))`
B. `(sqrtm-sqrt(m-1))/(sqrtm + sqrt(m+1))`
C. `(sqrtm + sqrt(m-1))/(sqrtm-sqrt(m-1))`
D. `(sqrtm - sqrt(m+1))/(sqrtm+sqrt(m+1))`

Answer

To solve the problem step by step, we will start with the definitions of the Arithmetic Mean (A.M) and Harmonic Mean (H.M) and then use the given condition to find the ratio \( a:b \).

### Step 1: Define A.M and H.M
The Arithmetic Mean (A.M) of two numbers \( a \) and \( b \) is given by:
\[ A.M = \frac{a + b}{2} \]
The Harmonic Mean (H.M) of two numbers \( a \) and \( b \) is given by:
\[ H.M = \frac{2ab}{a + b} \]

### Step 2: Set up the equation based on the problem statement
According to the problem, the A.M is \( m \) times the H.M. Therefore, we can write:
\[ \frac{a + b}{2} = m \cdot \frac{2ab}{a + b} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ (a + b)^2 = 4mab \]

### Step 4: Rearrange the equation
We can rearrange this equation to isolate the terms:
\[ (a + b)^2 - 4mab = 0 \]

### Step 5: Expand the left-hand side
Expanding \( (a + b)^2 \) gives:
\[ a^2 + 2ab + b^2 - 4mab = 0 \]
This simplifies to:
\[ a^2 + b^2 + 2ab - 4mab = 0 \]
or
\[ a^2 + b^2 - (4m - 2)ab = 0 \]

### Step 6: Factor the quadratic equation
This is a quadratic equation in terms of \( a \):
\[ a^2 - (4m - 2)ab + b^2 = 0 \]
Using the quadratic formula \( a = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \) where \( A = 1, B = -(4m - 2)b, C = b^2 \):
\[ a = \frac{(4m - 2)b \pm \sqrt{(4m - 2)^2b^2 - 4b^2}}{2} \]

### Step 7: Simplify the expression
This simplifies to:
\[ a = \frac{(4m - 2)b \pm b\sqrt{(4m - 2)^2 - 4}}{2} \]
Factoring out \( b \):
\[ a = b \cdot \frac{(4m - 2) \pm \sqrt{(4m - 2)^2 - 4}}{2} \]

### Step 8: Find the ratio \( a:b \)
Thus, the ratio \( \frac{a}{b} \) is:
\[ \frac{a}{b} = \frac{(4m - 2) \pm \sqrt{(4m - 2)^2 - 4}}{2} \]

### Step 9: Finalize the ratio
To find the ratio \( a:b \), we can express it as:
\[ a:b = (4m - 2) + \sqrt{(4m - 2)^2 - 4} : 2 \]
or
\[ a:b = (4m - 2) - \sqrt{(4m - 2)^2 - 4} : 2 \]