If `(1)/(a+b+7)=(1)/(a)+(1)/(b)+(1)/(7) and a+bne0 and a+bne-7`, then the possiblities are:-

Options

A. `a=7orb=7`
B. `a=7andb=-7`
C. `a=-7orb=7`
D. `a=-7orb=-7`

Answer

To solve the equation \(\frac{1}{a+b+7} = \frac{1}{a} + \frac{1}{b} + \frac{1}{7}\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \frac{1}{a+b+7} = \frac{1}{a} + \frac{1}{b} + \frac{1}{7} \]

### Step 2: Find a common denominator on the right side
The common denominator for the right side is \(ab \cdot 7\). Thus, we can rewrite the right side as:
\[ \frac{b \cdot 7 + a \cdot 7 + a \cdot b}{ab \cdot 7} \]
This simplifies to:
\[ \frac{7a + 7b + ab}{ab \cdot 7} \]

### Step 3: Set the two sides equal
Now we have:
\[ \frac{1}{a+b+7} = \frac{7a + 7b + ab}{ab \cdot 7} \]

### Step 4: Cross-multiply
Cross-multiplying gives us:
\[ ab \cdot 7 = (a+b+7)(7a + 7b + ab) \]

### Step 5: Expand the right side
Expanding the right side:
\[ ab \cdot 7 = (7a + 7b + ab)(a + b + 7) \]
This will lead to a polynomial equation in \(a\) and \(b\).

### Step 6: Analyze conditions
We know from the problem statement that \(a + b \neq 0\) and \(a + b \neq -7\). We will check the options given in the question to see which ones satisfy these conditions.

### Step 7: Check each option
1. **Option A**: \(a = 7\), \(b = 7\)
- \(a + b = 14 \neq 0\) and \(14 \neq -7\) (valid)

2. **Option B**: \(a = 7\), \(b = -7\)
- \(a + b = 0\) (invalid)

3. **Option C**: \(a = -7\), \(b = 7\)
- \(a + b = 0\) (invalid)

4. **Option D**: \(a = -7\), \(b = -7\)
- \(a + b = -14 \neq 0\) and \(-14 \neq -7\) (valid)

### Conclusion
The valid options are A and D. However, we need to find the correct answer based on the equation.

### Step 8: Substitute values to check
- For **Option A**:
\[ \frac{1}{7 + 7 + 7} = \frac{1}{21} \quad \text{and} \quad \frac{1}{7} + \frac{1}{7} + \frac{1}{7} = \frac{3}{7} \quad \text{(not equal)} \]

- For **Option D**:
\[ \frac{1}{-7 + -7 + 7} = \frac{1}{-7} \quad \text{and} \quad \frac{1}{-7} + \frac{1}{-7} + \frac{1}{7} = -\frac{1}{7} \quad \text{(equal)} \]

Thus, the correct answer is **Option D**: \(a = -7\) and \(b = -7\).