If `a+b+c =p, abc =q and ab+bc +ca =0` then what is the value of `a^(2) b^(2) +b^(2)c^(2) +c^(2)a^(2)`?

Options

A. 2pq
B. `-2pq`
C. 3pq
D. `-3pq`

Answer

To solve the problem, we need to find the value of \( a^2b^2 + b^2c^2 + c^2a^2 \) given the conditions \( a + b + c = p \), \( abc = q \), and \( ab + bc + ca = 0 \).

### Step-by-step Solution:

1. **Start with the identity**:
We know that:
\[ (ab + bc + ca)^2 = a^2b^2 + b^2c^2 + c^2a^2 + 2abc(a + b + c) \]
This identity relates the sum of products of the variables to the squares of the products.

2. **Substitute the known values**:
From the problem, we have \( ab + bc + ca = 0 \). Therefore:
\[ (ab + bc + ca)^2 = 0^2 = 0 \]
This simplifies our equation to:
\[ 0 = a^2b^2 + b^2c^2 + c^2a^2 + 2abc(a + b + c) \]

3. **Substitute \( a + b + c \) and \( abc \)**:
We can substitute \( a + b + c = p \) and \( abc = q \) into the equation:
\[ 0 = a^2b^2 + b^2c^2 + c^2a^2 + 2q(p) \]

4. **Rearranging the equation**:
Rearranging gives us:
\[ a^2b^2 + b^2c^2 + c^2a^2 = -2pq \]

5. **Final result**:
Thus, the value of \( a^2b^2 + b^2c^2 + c^2a^2 \) is:
\[ \boxed{-2pq} \]