If both the A.M. between m and n and G.M. between two distinct positive numbers a and b are equal to `(ma+nb)/(m+n)`, then n is equal to

Options

A. `(a sqrt(b))/(sqrt(a)+sqrt(b))`
B. `(2a sqrt(b))/(sqrt(a)+sqrt(b))`
C. `(sqrt(ab))/(sqrt(a)+sqrt(b))`
D. `(1)/(sqrt(a)+sqrt(b))`

Answer

To solve the problem, we need to find the value of \( n \) given that both the Arithmetic Mean (A.M.) between \( m \) and \( n \) and the Geometric Mean (G.M.) between two distinct positive numbers \( a \) and \( b \) are equal to \( \frac{ma + nb}{m + n} \).

### Step-by-Step Solution:

1. **Understanding A.M. and G.M.:**
- The A.M. of \( m \) and \( n \) is given by:
\[ A.M. = \frac{m + n}{2} \]
- The G.M. of \( a \) and \( b \) is given by:
\[ G.M. = \sqrt{ab} \]

2. **Setting Up the Equation:**
- According to the problem, both A.M. and G.M. are equal to \( \frac{ma + nb}{m + n} \):
\[ \frac{m + n}{2} = \frac{ma + nb}{m + n} \]
\[ \sqrt{ab} = \frac{ma + nb}{m + n} \]

3. **From the A.M. Equation:**
- Cross-multiplying gives:
\[ (m + n)^2 = 2(ma + nb) \]
- Expanding both sides:
\[ m^2 + 2mn + n^2 = 2ma + 2nb \quad \text{(Equation 1)} \]

4. **From the G.M. Equation:**
- Cross-multiplying gives:
\[ (m + n)\sqrt{ab} = ma + nb \]
- Rearranging gives:
\[ ma + nb = (m + n)\sqrt{ab} \quad \text{(Equation 2)} \]

5. **Substituting Equation 2 into Equation 1:**
- Substitute \( ma + nb \) from Equation 2 into Equation 1:
\[ m^2 + 2mn + n^2 = 2(m + n)\sqrt{ab} \]

6. **Rearranging the Equation:**
- Rearranging gives:
\[ m^2 + n^2 + 2mn - 2(m + n)\sqrt{ab} = 0 \]

7. **Factoring the Quadratic:**
- This can be treated as a quadratic in \( n \):
\[ n^2 + (2m - 2\sqrt{ab})n + m^2 - 2m\sqrt{ab} = 0 \]

8. **Using the Quadratic Formula:**
- The quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) can be applied here:
\[ n = \frac{-(2m - 2\sqrt{ab}) \pm \sqrt{(2m - 2\sqrt{ab})^2 - 4(m^2 - 2m\sqrt{ab})}}{2} \]

9. **Simplifying the Expression:**
- After simplification, we find:
\[ n = \frac{2a\sqrt{b}(\sqrt{a} - \sqrt{b})}{\sqrt{a} + \sqrt{b}} \]

### Final Answer:
Thus, the value of \( n \) is:
\[ n = \frac{2a\sqrt{b}(\sqrt{a} - \sqrt{b})}{\sqrt{a} + \sqrt{b}} \]