If ` a + b + c = 0` then the value of ` (a ^(2) + b^(2) + c^(2))/( a^(2) - ba)` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we start with the given equation:

**Given:**
\( a + b + c = 0 \)

We need to find the value of:
\[ \frac{a^2 + b^2 + c^2}{a^2 - ba} \]

### Step 1: Express \(c\) in terms of \(a\) and \(b\)
From the equation \(a + b + c = 0\), we can express \(c\) as:
\[ c = - (a + b) \]

### Step 2: Substitute \(c\) into the expression \(a^2 + b^2 + c^2\)
Now we substitute \(c\) into the expression \(a^2 + b^2 + c^2\):
\[ c^2 = (- (a + b))^2 = (a + b)^2 = a^2 + 2ab + b^2 \]
Thus,
\[ a^2 + b^2 + c^2 = a^2 + b^2 + (a^2 + 2ab + b^2) = 2a^2 + 2b^2 + 2ab \]
This simplifies to:
\[ a^2 + b^2 + c^2 = 2(a^2 + ab + b^2) \]

### Step 3: Substitute \(c\) into the denominator \(a^2 - ba\)
Now we substitute \(c\) into the denominator \(a^2 - ba\):
\[ a^2 - ba = a^2 - b \cdot a \]

### Step 4: Substitute into the entire expression
Now we can substitute our results back into the expression:
\[ \frac{a^2 + b^2 + c^2}{a^2 - ba} = \frac{2(a^2 + ab + b^2)}{a^2 - ba} \]

### Step 5: Simplify the expression
Now we can simplify the expression:
\[ = \frac{2(a^2 + ab + b^2)}{a^2 - ab} \]

### Step 6: Factor out common terms if possible
Notice that \(a^2 - ab\) can be factored:
\[ = \frac{2(a^2 + ab + b^2)}{a(a - b)} \]

### Step 7: Evaluate the expression
Now, we can evaluate the expression. However, without specific values for \(a\) and \(b\), we can only express it in this simplified form.

### Final Result
Thus, the value of the expression is:
\[ \frac{2(a^2 + ab + b^2)}{a(a - b)} \]