In sub - part (i) to (x) choose the correct option and in sub - part (xi) to (xy), answer the questions as intructed .
If A is the A.M between a and b , then `(A+2a)/(A-b)+(A+2b)/(A-a)=`

Options

A. 2
B. 0
C. `-4`
D. 4

Answer

To solve the problem, we need to find the value of the expression \((A + 2a)/(A - b) + (A + 2b)/(A - a)\) given that \(A\) is the arithmetic mean (A.M.) of \(a\) and \(b\).

### Step-by-step Solution:

1. **Understanding the Arithmetic Mean**:
The arithmetic mean \(A\) of two numbers \(a\) and \(b\) is given by:
\[ A = \frac{a + b}{2} \]

2. **Substituting \(A\) in the Expression**:
We substitute \(A\) into the expression:
\[ \frac{A + 2a}{A - b} + \frac{A + 2b}{A - a} \]
becomes:
\[ \frac{\frac{a + b}{2} + 2a}{\frac{a + b}{2} - b} + \frac{\frac{a + b}{2} + 2b}{\frac{a + b}{2} - a} \]

3. **Simplifying the First Fraction**:
For the first term:
\[ \frac{\frac{a + b + 4a}{2}}{\frac{a + b - 2b}{2}} = \frac{\frac{5a + b}{2}}{\frac{a - b}{2}} = \frac{5a + b}{a - b} \]

4. **Simplifying the Second Fraction**:
For the second term:
\[ \frac{\frac{a + b + 4b}{2}}{\frac{a + b - 2a}{2}} = \frac{\frac{a + 5b}{2}}{\frac{-a + b}{2}} = \frac{a + 5b}{-a + b} = -\frac{a + 5b}{a - b} \]

5. **Combining the Two Fractions**:
Now, we combine the two fractions:
\[ \frac{5a + b}{a - b} - \frac{a + 5b}{a - b} = \frac{(5a + b) - (a + 5b)}{a - b} \]
Simplifying the numerator:
\[ (5a + b - a - 5b) = 4a - 4b = 4(a - b) \]

6. **Final Simplification**:
Therefore, the expression simplifies to:
\[ \frac{4(a - b)}{a - b} = 4 \]

### Final Answer:
The value of the expression is:
\[ \boxed{4} \]