The A.M. between m and n and the G.M. between a and b are each equal to `(ma+nb)/(m+n)` . Then m=

Options

A. `(asqrt(b))/(sqrt(a)+sqrt(b))`
B. `(bsqrt(a))/(sqrt(a)+sqrt(b))`
C. `(2asqrt(b))/(sqrt(a)+sqrt(b))`
D. `(2bsqrt(a))/(sqrt(a)+sqrt(b))`

Answer

To solve the problem, we need to find the value of \( m \) given that the arithmetic mean (A.M.) between \( m \) and \( n \) and the geometric mean (G.M.) between \( a \) and \( b \) are both equal to \( \frac{ma + nb}{m + n} \).

### Step-by-Step Solution

1. **Write the expression for the A.M.**
The arithmetic mean (A.M.) of \( m \) and \( n \) is given by:
\[ \text{A.M.} = \frac{m + n}{2} \]

2. **Set the A.M. equal to the given expression.**
According to the problem, we have:
\[ \frac{m + n}{2} = \frac{ma + nb}{m + n} \]

3. **Cross-multiply to eliminate the fraction.**
Cross-multiplying gives us:
\[ (m + n)(m + n) = 2(ma + nb) \]
Simplifying this, we get:
\[ (m + n)^2 = 2(ma + nb) \]

4. **Expand both sides.**
Expanding the left side:
\[ m^2 + 2mn + n^2 = 2ma + 2nb \]

5. **Rearrange the equation.**
Rearranging gives:
\[ m^2 + 2mn + n^2 - 2ma - 2nb = 0 \]

6. **Write the expression for the G.M.**
The geometric mean (G.M.) of \( a \) and \( b \) is given by:
\[ \text{G.M.} = \sqrt{ab} \]

7. **Set the G.M. equal to the given expression.**
According to the problem, we have:
\[ \sqrt{ab} = \frac{ma + nb}{m + n} \]

8. **Cross-multiply to eliminate the fraction.**
Cross-multiplying gives us:
\[ \sqrt{ab}(m + n) = ma + nb \]

9. **Rearrange the equation.**
Rearranging gives:
\[ ma + nb - m\sqrt{ab} - n\sqrt{ab} = 0 \]

10. **Combine the two equations.**
We now have two equations:
- From A.M.: \( m^2 + 2mn + n^2 - 2ma - 2nb = 0 \)
- From G.M.: \( ma + nb - m\sqrt{ab} - n\sqrt{ab} = 0 \)

11. **Solve for \( m \).**
By manipulating these equations, we can find \( m \) in terms of \( a \) and \( b \).

From the G.M. equation, we can express \( ma + nb \) as:
\[ ma + nb = (m + n)\sqrt{ab} \]

Substitute this back into the A.M. equation:
\[ m^2 + 2mn + n^2 - 2(m + n)\sqrt{ab} = 0 \]

12. **Factor or use the quadratic formula.**
This is a quadratic equation in terms of \( m \). We can solve for \( m \) using the quadratic formula or by factoring.

13. **Final expression for \( m \).**
After simplification, we find:
\[ m = \frac{2b(a - \sqrt{ab})}{a - b} \]