If the sum of the roots of equation `(1)/(x+a)+(1)/(x+b)=1/c` is zero, then the product of roots is :

Options

A. 0
B. `(a+b)/(2)`
C. `-(1)/(2)(a^(2)+b^(2))`
D. `2(a^(2)+b^(2))`

Answer

To solve the equation \( \frac{1}{x+a} + \frac{1}{x+b} = \frac{1}{c} \) and find the product of the roots given that the sum of the roots is zero, we can follow these steps:

### Step 1: Rewrite the Equation
Start with the equation:
\[ \frac{1}{x+a} + \frac{1}{x+b} = \frac{1}{c} \]
To combine the left-hand side, we need a common denominator:
\[ \frac{(x+b) + (x+a)}{(x+a)(x+b)} = \frac{1}{c} \]
This simplifies to:
\[ \frac{2x + (a+b)}{(x+a)(x+b)} = \frac{1}{c} \]

### Step 2: Cross-Multiply
Cross-multiplying gives:
\[ c(2x + (a+b)) = (x+a)(x+b) \]

### Step 3: Expand Both Sides
Expanding the right-hand side:
\[ (x+a)(x+b) = x^2 + (a+b)x + ab \]
Thus, we have:
\[ c(2x + (a+b)) = x^2 + (a+b)x + ab \]

### Step 4: Rearrange to Form a Quadratic Equation
Rearranging the equation gives:
\[ x^2 + (a+b - 2c)x + (ab - c(a+b)) = 0 \]
This is a quadratic equation in the standard form \( Ax^2 + Bx + C = 0 \) where:
- \( A = 1 \)
- \( B = a + b - 2c \)
- \( C = ab - c(a+b) \)

### Step 5: Use the Given Condition
We know the sum of the roots \( \alpha + \beta = 0 \). From the quadratic formula, the sum of the roots is given by:
\[ -\frac{B}{A} = 0 \]
This implies:
\[ B = a + b - 2c = 0 \]
Thus, we can solve for \( c \):
\[ 2c = a + b \quad \Rightarrow \quad c = \frac{a + b}{2} \]

### Step 6: Find the Product of the Roots
The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{C}{A} = ab - c(a+b) \]
Substituting \( c = \frac{a + b}{2} \):
\[ \alpha \beta = ab - \frac{a + b}{2}(a + b) \]
This simplifies to:
\[ \alpha \beta = ab - \frac{(a+b)^2}{2} \]

### Step 7: Further Simplification
Now we can express \( \alpha \beta \):
\[ \alpha \beta = ab - \frac{a^2 + 2ab + b^2}{2} = ab - \frac{a^2 + b^2 + 2ab}{2} \]
This leads to:
\[ \alpha \beta = ab - \left(\frac{a^2 + b^2}{2} + ab\right) = -\frac{a^2 + b^2}{2} \]

### Final Answer
Thus, the product of the roots is:
\[ \alpha \beta = -\frac{a^2 + b^2}{2} \]