For some x and y that satisfy the equation `xy=- x ^(2)` , which of the following is FALSE ?

Options

A. `x((1)/(y))=-1`
B. `x^(2)((1)/(Y^(2)))=1`
C. `x^(2) +y^(2) =-2xy`
D. `x^(3) -y^(3)=0`

Answer

To solve the problem, we need to analyze the equation \(xy = -x^2\) and check which of the given options is false.

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ xy = -x^2 \]

2. **Rearrange the equation to express \(y\) in terms of \(x\)**:
\[ y = -\frac{x^2}{x} \quad \text{(assuming \(x \neq 0\))} \]
Simplifying gives:
\[ y = -x \]

3. **Now, we will check each option one by one to see if they satisfy the equation**.

**Option A**: \(x \cdot \frac{1}{y} = -1\)

Substitute \(y = -x\):
\[ x \cdot \frac{1}{-x} = -1 \]
Simplifying gives:
\[ -1 = -1 \quad \text{(True)} \]

**Option B**: \(x^2 \cdot \frac{1}{y^2} = 1\)

Substitute \(y = -x\):
\[ x^2 \cdot \frac{1}{(-x)^2} = 1 \]
Simplifying gives:
\[ x^2 \cdot \frac{1}{x^2} = 1 \quad \text{(True)} \]

**Option C**: \(x^2 + y^2 = -2xy\)

Substitute \(y = -x\):
\[ x^2 + (-x)^2 = -2x(-x) \]
Simplifying gives:
\[ x^2 + x^2 = 2x^2 \]
This simplifies to:
\[ 2x^2 = 2x^2 \quad \text{(True)} \]

**Option D**: \(x^3 - y^3 = 0\)

Substitute \(y = -x\):
\[ x^3 - (-x)^3 = 0 \]
Simplifying gives:
\[ x^3 - (-x^3) = 0 \]
This simplifies to:
\[ x^3 + x^3 = 0 \quad \Rightarrow \quad 2x^3 = 0 \]
This is only true if \(x = 0\). For other values of \(x\), it does not hold true.

4. **Conclusion**: The option that is false is:
\[ \text{Option D: } x^3 - y^3 = 0 \]