If `(a ^(2) - b c)/( a ^(2 ) + bc)+ (b ^(2) -ca)/( b ^(2) + ca) + (c ^(2) - ab)/( c ^(2) + ab) =1,` then the value of `(a ^(2))/(a ^(2) + bc ) + (b ^(2))/(b ^(2) + ac) + (c ^(2))/( c ^(2) + ab)` is

Options

A. `-1`
B. `2`
C. 0
D. 1

Answer

To solve the equation

\[ \frac{a^2 - bc}{a^2 + bc} + \frac{b^2 - ca}{b^2 + ca} + \frac{c^2 - ab}{c^2 + ab} = 1, \]

we can start by rewriting each term in the left-hand side.

### Step 1: Rewrite each term
We can express each fraction in a different form:

\[ \frac{a^2 - bc}{a^2 + bc} = 1 - \frac{2bc}{a^2 + bc}, \]
\[ \frac{b^2 - ca}{b^2 + ca} = 1 - \frac{2ca}{b^2 + ca}, \]
\[ \frac{c^2 - ab}{c^2 + ab} = 1 - \frac{2ab}{c^2 + ab}. \]

### Step 2: Substitute back into the equation
Substituting these into the original equation gives:

\[ \left(1 - \frac{2bc}{a^2 + bc}\right) + \left(1 - \frac{2ca}{b^2 + ca}\right) + \left(1 - \frac{2ab}{c^2 + ab}\right) = 1. \]

### Step 3: Simplify the equation
This simplifies to:

\[ 3 - \left(\frac{2bc}{a^2 + bc} + \frac{2ca}{b^2 + ca} + \frac{2ab}{c^2 + ab}\right) = 1. \]

### Step 4: Isolate the sum of fractions
Rearranging gives:

\[ \frac{2bc}{a^2 + bc} + \frac{2ca}{b^2 + ca} + \frac{2ab}{c^2 + ab} = 2. \]

### Step 5: Divide by 2
Dividing the entire equation by 2 gives:

\[ \frac{bc}{a^2 + bc} + \frac{ca}{b^2 + ca} + \frac{ab}{c^2 + ab} = 1. \]

### Step 6: Find the desired expression
Now we need to find the value of:

\[ \frac{a^2}{a^2 + bc} + \frac{b^2}{b^2 + ca} + \frac{c^2}{c^2 + ab}. \]

### Step 7: Rewrite the expression
We can rewrite each term in the desired expression as follows:

\[ \frac{a^2}{a^2 + bc} = 1 - \frac{bc}{a^2 + bc}, \]
\[ \frac{b^2}{b^2 + ca} = 1 - \frac{ca}{b^2 + ca}, \]
\[ \frac{c^2}{c^2 + ab} = 1 - \frac{ab}{c^2 + ab}. \]

### Step 8: Substitute back into the expression
Substituting these into the desired expression gives:

\[ \left(1 - \frac{bc}{a^2 + bc}\right) + \left(1 - \frac{ca}{b^2 + ca}\right) + \left(1 - \frac{ab}{c^2 + ab}\right). \]

### Step 9: Simplify the expression
This simplifies to:

\[ 3 - \left(\frac{bc}{a^2 + bc} + \frac{ca}{b^2 + ca} + \frac{ab}{c^2 + ab}\right). \]

### Step 10: Substitute the known value
Since we found that

\[ \frac{bc}{a^2 + bc} + \frac{ca}{b^2 + ca} + \frac{ab}{c^2 + ab} = 1, \]

we can substitute this into our expression:

\[ 3 - 1 = 2. \]

### Conclusion
Thus, the value of

\[ \frac{a^2}{a^2 + bc} + \frac{b^2}{b^2 + ca} + \frac{c^2}{c^2 + ab} = 2. \]