If `x+y+z =10 , x^(2) +y^(2) +z^(2) =60` then value of `xy+yz +zx` is

Options

A. 40
B. 80
C. 160
D. 20

Answer

To find the value of \( xy + yz + zx \) given that \( x + y + z = 10 \) and \( x^2 + y^2 + z^2 = 60 \), we can use the algebraic identity for the square of a sum.

### Step-by-Step Solution:

1. **Start with the given equations:**
\[ x + y + z = 10 \]
\[ x^2 + y^2 + z^2 = 60 \]

2. **Use the identity for the square of a sum:**
The identity states:
\[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx) \]

3. **Substitute the known values into the identity:**
From the first equation, we know:
\[ (x + y + z)^2 = 10^2 = 100 \]
Therefore, we can write:
\[ 100 = x^2 + y^2 + z^2 + 2(xy + yz + zx) \]

4. **Substitute \( x^2 + y^2 + z^2 \) into the equation:**
We know \( x^2 + y^2 + z^2 = 60 \), so we substitute this into the equation:
\[ 100 = 60 + 2(xy + yz + zx) \]

5. **Solve for \( 2(xy + yz + zx) \):**
Rearranging the equation gives:
\[ 100 - 60 = 2(xy + yz + zx) \]
\[ 40 = 2(xy + yz + zx) \]

6. **Divide by 2 to find \( xy + yz + zx \):**
\[ xy + yz + zx = \frac{40}{2} = 20 \]

### Final Answer:
The value of \( xy + yz + zx \) is \( 20 \).

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