If ` x + y + z = 0` ,then the value of ` ( x^(2) + y^(2) + z^(2))/( x^(2) - yz)` is

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To solve the problem, we need to find the value of the expression \(\frac{x^2 + y^2 + z^2}{x^2 - yz}\) given that \(x + y + z = 0\).

### Step-by-step Solution:

1. **Use the given equation**:
We know that \(x + y + z = 0\). This can be rearranged to express \(z\) in terms of \(x\) and \(y\):
\[ z = - (x + y) \]

2. **Substitute \(z\) into the expression**:
We need to find \(x^2 + y^2 + z^2\) and \(x^2 - yz\). First, we will calculate \(z^2\):
\[ z^2 = (- (x + y))^2 = (x + y)^2 = x^2 + 2xy + y^2 \]
Now substituting \(z^2\) into \(x^2 + y^2 + z^2\):
\[ x^2 + y^2 + z^2 = x^2 + y^2 + (x^2 + 2xy + y^2) = 2x^2 + 2y^2 + 2xy = 2(x^2 + y^2 + xy) \]

3. **Calculate \(yz\)**:
Next, we calculate \(yz\):
\[ yz = y \cdot z = y \cdot (- (x + y)) = -y(x + y) = -yx - y^2 \]
Now substituting this into \(x^2 - yz\):
\[ x^2 - yz = x^2 + yx + y^2 \]

4. **Combine the results**:
Now we can substitute both results into our original expression:
\[ \frac{x^2 + y^2 + z^2}{x^2 - yz} = \frac{2(x^2 + y^2 + xy)}{x^2 + yx + y^2} \]

5. **Simplify the expression**:
Notice that \(x^2 + yx + y^2\) can be factored or simplified. However, we can also evaluate this expression by substituting specific values for \(x\) and \(y\) that satisfy \(x + y + z = 0\). For simplicity, let's set \(y = 0\) and \(z = -x\):
- Then, \(x + 0 - x = 0\) holds true.
- Now substituting \(y = 0\) into our expression:
\[ \frac{2(x^2 + 0 + 0)}{x^2 + 0 + 0} = \frac{2x^2}{x^2} = 2 \]

Thus, the value of the expression \(\frac{x^2 + y^2 + z^2}{x^2 - yz}\) is **2**.