If `(a+b+c)=6 and a^(2) +b^(2)+c^(2)=14`, then `(ab+bc+ca)=`?
A. 22
B. 11
C. 33
D. 44

Options

A. B
B. D
C. A
D. C

Answer

To solve the problem, we need to find the value of \( ab + bc + ca \) given that \( a + b + c = 6 \) and \( a^2 + b^2 + c^2 = 14 \).

### Step-by-step Solution:

1. **Use the identity for the square of a sum**:
The formula for the square of a sum is:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]

2. **Substitute the known values**:
We know that \( a + b + c = 6 \) and \( a^2 + b^2 + c^2 = 14 \). Plugging these values into the formula gives:
\[ (6)^2 = 14 + 2(ab + bc + ca) \]

3. **Calculate \( (6)^2 \)**:
\[ 36 = 14 + 2(ab + bc + ca) \]

4. **Rearrange the equation**:
To isolate \( 2(ab + bc + ca) \), subtract 14 from both sides:
\[ 36 - 14 = 2(ab + bc + ca) \]
\[ 22 = 2(ab + bc + ca) \]

5. **Solve for \( ab + bc + ca \)**:
Divide both sides by 2:
\[ ab + bc + ca = \frac{22}{2} = 11 \]

### Final Answer:
Thus, the value of \( ab + bc + ca \) is \( 11 \).