`Delta=|{:(1+a^2+a^4,1+ab+a^2b^2,1+ac+a^2c^2),(1+ab+a^2b^2,1+b^2+b^4,1+bc+b^2c^2),(1+ac+a^2c^2,1+bc+b^2c^2,1+c^2+c^4):}|` is equal to

Options

A. `(a+b+c)^6`
B. `(a-b)^2(b-c)^2(c-a)^2`
C. `4(a-b)(b-c)(c-a)`
D. none of these

Answer

To solve the determinant given by

\[ \Delta = \begin{vmatrix} 1 + a^2 + a^4 & 1 + ab + a^2b^2 & 1 + ac + a^2c^2 \\ 1 + ab + a^2b^2 & 1 + b^2 + b^4 & 1 + bc + b^2c^2 \\ 1 + ac + a^2c^2 & 1 + bc + b^2c^2 & 1 + c^2 + c^4 \end{vmatrix} \]

we will compute the determinant step by step.

### Step 1: Simplifying the Determinant
Notice that the elements of the determinant can be expressed in terms of squares. For instance, we can rewrite the first row as:

- \(1 + a^2 + a^4 = (a^2 + 1)^2\)
- \(1 + ab + a^2b^2 = (ab + 1)^2\)
- \(1 + ac + a^2c^2 = (ac + 1)^2\)

Thus, we can rewrite the determinant as:

\[ \Delta = \begin{vmatrix} (a^2 + 1)^2 & (ab + 1)^2 & (ac + 1)^2 \\ (ab + 1)^2 & (b^2 + 1)^2 & (bc + 1)^2 \\ (ac + 1)^2 & (bc + 1)^2 & (c^2 + 1)^2 \end{vmatrix} \]

### Step 2: Applying the Determinant Properties
Using the property of determinants that states if two rows (or columns) are proportional, the determinant is zero. We can check if any two rows are proportional.

### Step 3: Checking for Proportional Rows
We can see that if we take the first row and the second row, they are not proportional since they contain different variables. However, we can check if there is any linear combination that can yield zero.

### Step 4: Evaluating the Determinant
To evaluate the determinant, we can use expansion by minors or row operations to simplify it. However, given the symmetry and structure of the determinant, we can also use the fact that if we substitute \(a = b = c\), the determinant simplifies significantly.

Let \(a = b = c = k\):

\[ \Delta = \begin{vmatrix} (1 + k^2)^2 & (k^2 + 1)^2 & (k^2 + 1)^2 \\ (k^2 + 1)^2 & (1 + k^2)^2 & (k^2 + 1)^2 \\ (k^2 + 1)^2 & (k^2 + 1)^2 & (1 + k^2)^2 \end{vmatrix} \]

This determinant will yield zero due to the repetition of rows.

### Conclusion
Thus, we conclude that:

\[ \Delta = 0 \]