If ` x + y + z = 0 ` then what is the value of ` ( xy + yz + zx)/( x^(2) + y^(2) + z^(2))`

Options

A. 1
B. `-1`
C. `(1)/(2)`
D. `-(1)/(2)`

Answer

To solve the problem, we start with the equation given:

1. **Given**: \( x + y + z = 0 \)

We need to find the value of:

\[ \frac{xy + yz + zx}{x^2 + y^2 + z^2} \]

2. **Square the equation**: We square both sides of the equation \( x + y + z = 0 \):

\[ (x + y + z)^2 = 0^2 \]

This expands to:

\[ x^2 + y^2 + z^2 + 2(xy + yz + zx) = 0 \]

3. **Rearranging the equation**: We can rearrange this equation to isolate \( x^2 + y^2 + z^2 \):

\[ x^2 + y^2 + z^2 = -2(xy + yz + zx) \]

4. **Substituting into the fraction**: Now we substitute this back into our original expression:

\[ \frac{xy + yz + zx}{x^2 + y^2 + z^2} = \frac{xy + yz + zx}{-2(xy + yz + zx)} \]

5. **Simplifying the fraction**: Assuming \( xy + yz + zx \neq 0 \), we can simplify:

\[ \frac{xy + yz + zx}{-2(xy + yz + zx)} = \frac{1}{-2} = -\frac{1}{2} \]

Thus, the value of \( \frac{xy + yz + zx}{x^2 + y^2 + z^2} \) is:

\[ \boxed{-\frac{1}{2}} \]