If `Delta^2=|(b^2+c^2,ab,ac),(ab,c^2+a^2,bc),(ac,bc,a^2+b^2)|` , then `Delta` is equal to

Options

A. `|(0,c,b),(c,0,a),(b,a,0)|`
B. `|(a,b,c),(b,c,a),(c,a,b)|`
C. `|(0,-c,b),(c,0,-a),(-b,-a,0)|`
D. none

Answer

To solve the problem, we need to find the value of \( \Delta \) given that \( \Delta^2 = \begin{vmatrix} b^2 + c^2 & ab & ac \\ ab & c^2 + a^2 & bc \\ ac & bc & a^2 + b^2 \end{vmatrix} \).

### Step-by-Step Solution:

1. **Identify the Determinant**:
We start with the determinant given in the problem:
\[ \Delta^2 = \begin{vmatrix} b^2 + c^2 & ab & ac \\ ab & c^2 + a^2 & bc \\ ac & bc & a^2 + b^2 \end{vmatrix} \]

2. **Use Properties of Determinants**:
We can express the determinant as a product of two identical determinants. This is a common technique in determinant problems. We can write:
\[ \Delta^2 = \begin{vmatrix} b & c & 0 \\ 0 & a & b \\ 0 & 0 & c \end{vmatrix} \times \begin{vmatrix} b & c & 0 \\ 0 & a & b \\ 0 & 0 & c \end{vmatrix} \]

3. **Calculate the Determinant**:
Let's calculate the determinant:
\[ \Delta^2 = \begin{vmatrix} b^2 + c^2 & ab & ac \\ ab & c^2 + a^2 & bc \\ ac & bc & a^2 + b^2 \end{vmatrix} \]
We can expand this determinant using cofactor expansion or any other method.

4. **Simplify the Determinant**:
After performing the calculations (using row operations or cofactor expansion), we find that:
\[ \Delta^2 = (a^2 + b^2 + c^2)^2 \]

5. **Take the Square Root**:
To find \( \Delta \), we take the square root of both sides:
\[ \Delta = a^2 + b^2 + c^2 \]

### Final Result:
Thus, the value of \( \Delta \) is:
\[ \Delta = a^2 + b^2 + c^2 \]