If `a, b and c` are distinct positive real numbers such that `Delta_(1) = |(a,b,c),(b,c,a),(c,a,b)|` and `Delta_(2) = |(bc - a^2, ac -b^2, ab - c^2),(ac - b^2, ab - c^2, bc -a^2),(ab -c^2, bc - a^2, ac - b^2)|`, then

Options

A. `Delta_(1) = Delta_(2)`
B. `Delta_(1)^(2) + Delta_(2) = 0`
C. `Delta_(1)^(2) = Delta_(2)`
D. `Delta_(1)^(2) = Delta_(2)^(2)`

Answer

To solve the problem, we need to analyze the determinants \( \Delta_1 \) and \( \Delta_2 \) given in the question.

### Step 1: Calculate \( \Delta_1 \)

The determinant \( \Delta_1 \) is given by:

\[ \Delta_1 = \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} \]

To compute this determinant, we can use the rule of Sarrus or cofactor expansion. Let's use the cofactor expansion along the first row:

\[ \Delta_1 = a \begin{vmatrix} c & a \\ a & b \end{vmatrix} - b \begin{vmatrix} b & a \\ c & b \end{vmatrix} + c \begin{vmatrix} b & c \\ c & a \end{vmatrix} \]

Calculating each of these 2x2 determinants:

1. \( \begin{vmatrix} c & a \\ a & b \end{vmatrix} = cb - a^2 \)

2. \( \begin{vmatrix} b & a \\ c & b \end{vmatrix} = bb - ac = b^2 - ac \)

3. \( \begin{vmatrix} b & c \\ c & a \end{vmatrix} = ba - c^2 \)

Substituting these back into the expression for \( \Delta_1 \):

\[ \Delta_1 = a(cb - a^2) - b(b^2 - ac) + c(ba - c^2) \]

### Step 2: Simplify \( \Delta_1 \)

Now, we simplify \( \Delta_1 \):

\[ \Delta_1 = acb - a^3 - b^3 + abc + abc - c^3 \]
\[ = 3abc - (a^3 + b^3 + c^3) \]

### Step 3: Calculate \( \Delta_2 \)

Next, we compute \( \Delta_2 \):

\[ \Delta_2 = \begin{vmatrix} bc - a^2 & ac - b^2 & ab - c^2 \\ ac - b^2 & ab - c^2 & bc - a^2 \\ ab - c^2 & bc - a^2 & ac - b^2 \end{vmatrix} \]

This determinant can also be computed using cofactor expansion or properties of determinants.

### Step 4: Relate \( \Delta_1 \) and \( \Delta_2 \)

From the properties of determinants, specifically the cofactor expansion, we can show that the determinant of the cofactor matrix of \( \Delta_1 \) is equal to \( \Delta_2 \).

### Conclusion

After analyzing both determinants, we find that:

\[ \Delta_1^2 = \Delta_2 \]

Thus, the correct option is:

**Option C: \( \Delta_1^2 = \Delta_2 \)**