The value of m for which the equation `(a)/(x+a+m)+(b)/(x+b+m)=1` has roots equal in magnitude but opposite in sign is :

Options

A. `(a-b)/(a+b)`
B. `-1`
C. 0
D. `(a+b)/(a-b)`

Answer

To solve the equation \(\frac{a}{x + a + m} + \frac{b}{x + b + m} = 1\) for the value of \(m\) such that the roots are equal in magnitude but opposite in sign, we can follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ \frac{a}{x + a + m} + \frac{b}{x + b + m} = 1 \]

### Step 2: Combine the fractions
To combine the fractions, find a common denominator:
\[ \frac{a(x + b + m) + b(x + a + m)}{(x + a + m)(x + b + m)} = 1 \]

### Step 3: Expand the numerator
Expand the numerator:
\[ a(x + b + m) + b(x + a + m) = ax + ab + am + bx + ba + bm \]
This simplifies to:
\[ (a + b)x + (ab + am + bm) \]

### Step 4: Set up the equation
Now we set the combined expression equal to the right side:
\[ (a + b)x + (ab + am + bm) = (x + a + m)(x + b + m) \]

### Step 5: Expand the right side
Expand the right side:
\[ (x + a + m)(x + b + m) = x^2 + (a + b + 2m)x + (ab + am + bm) \]

### Step 6: Equate coefficients
Now we can equate the coefficients from both sides. The left side has:
- Coefficient of \(x\): \(a + b\)
- Constant term: \(ab + am + bm\)

The right side has:
- Coefficient of \(x\): \(a + b + 2m\)
- Constant term: \(ab + am + bm\)

### Step 7: Set the coefficients equal
From the coefficients of \(x\):
\[ a + b = a + b + 2m \]
This simplifies to:
\[ 0 = 2m \implies m = 0 \]

### Step 8: Verify the condition for roots
Since we need the roots to be equal in magnitude but opposite in sign, we check:
If \(m = 0\), the equation simplifies to:
\[ \frac{a}{x + a} + \frac{b}{x + b} = 1 \]
This will yield roots that are equal in magnitude but opposite in sign.

### Final Answer
Thus, the value of \(m\) for which the equation has roots equal in magnitude but opposite in sign is:
\[ \boxed{0} \]