If ` (m -a^2)/( b^2 +c^2 ) + (m-b^2)/( c^2 +a^2) + (m-c ^2 )/( a^2 +b^2 )=3` , then the value of m is

Options

A. `a^2 +b^2 -c^2`
B. `a^2 +b^2`
C. `a^2 +b^2 +c^2`
D. `a^2 -b^2 -c^2`

Answer

To solve the equation

\[ \frac{m - a^2}{b^2 + c^2} + \frac{m - b^2}{c^2 + a^2} + \frac{m - c^2}{a^2 + b^2} = 3, \]

we can use the symmetry of the variables \(a\), \(b\), and \(c\).

### Step-by-Step Solution:

1. **Assume Symmetry**: Since the equation is symmetric in \(a\), \(b\), and \(c\), we can set \(a = b = c\). Let \(a = b = c = k\).

2. **Substitute Values**: Substitute \(a\), \(b\), and \(c\) with \(k\) in the equation:
\[ \frac{m - k^2}{k^2 + k^2} + \frac{m - k^2}{k^2 + k^2} + \frac{m - k^2}{k^2 + k^2} = 3. \]
This simplifies to:
\[ 3 \cdot \frac{m - k^2}{2k^2} = 3. \]

3. **Simplify the Equation**: We can simplify the left-hand side:
\[ \frac{3(m - k^2)}{2k^2} = 3. \]

4. **Clear the Denominator**: Multiply both sides by \(2k^2\):
\[ 3(m - k^2) = 6k^2. \]

5. **Distribute**: Distributing the \(3\) gives:
\[ 3m - 3k^2 = 6k^2. \]

6. **Rearrange the Equation**: Move \(3k^2\) to the other side:
\[ 3m = 6k^2 + 3k^2. \]
This simplifies to:
\[ 3m = 9k^2. \]

7. **Solve for \(m\)**: Divide both sides by \(3\):
\[ m = 3k^2. \]

### Conclusion:
Thus, the value of \(m\) is \(3a^2\) (since \(k\) was originally set to \(a\)).