If a,b, are three non-zero real numbers such that `a + b + c =0, and b ^(2) ne ` ca, then the value of ` (a ^(2) + b ^(2) + c ^(2))/( b ^(2) - ca )` is

Options

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

Answer

To solve the problem step by step, we start with the given equations and manipulate them to find the required value.

### Step 1: Write down the given equation
We are given that:
\[ a + b + c = 0 \]

### Step 2: Express one variable in terms of the others
From the equation \( a + b + c = 0 \), we can express \( c \) in terms of \( a \) and \( b \):
\[ c = - (a + b) \]

### Step 3: Substitute \( c \) into the expression we need to evaluate
We need to evaluate:
\[ \frac{a^2 + b^2 + c^2}{b^2 - ca} \]
Substituting \( c \) gives:
\[ \frac{a^2 + b^2 + (- (a + b))^2}{b^2 - (- (a + b))a} \]

### Step 4: Simplify the numerator
Now, simplify the numerator:
\[ c^2 = (- (a + b))^2 = (a + b)^2 = a^2 + 2ab + b^2 \]
Thus, the numerator becomes:
\[ a^2 + b^2 + c^2 = a^2 + b^2 + (a^2 + 2ab + b^2) = 2a^2 + 2b^2 + 2ab \]
So, the numerator simplifies to:
\[ 2(a^2 + ab + b^2) \]

### Step 5: Simplify the denominator
Now, simplify the denominator:
\[ b^2 - ca = b^2 - (- (a + b))a = b^2 + a(a + b) = b^2 + a^2 + ab \]

### Step 6: Substitute back into the expression
Now, we substitute these simplified forms back into our expression:
\[ \frac{2(a^2 + ab + b^2)}{b^2 + a^2 + ab} \]

### Step 7: Simplify the fraction
Notice that \( a^2 + ab + b^2 \) appears in both the numerator and the denominator:
\[ \frac{2(a^2 + ab + b^2)}{a^2 + ab + b^2} = 2 \]

### Conclusion
Thus, the value of the expression is:
\[ \boxed{2} \]