If `(a)/(b) + (b)/(a) = - 1,`then the value of `(a ^(2))/(b) - (b ^(2))/(a ) ` is

Options

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

Answer

To solve the equation \(\frac{a}{b} + \frac{b}{a} = -1\) and find the value of \(\frac{a^2}{b} - \frac{b^2}{a}\), we can follow these steps:

### Step 1: Rewrite the given equation
Start with the equation:
\[ \frac{a}{b} + \frac{b}{a} = -1 \]

### Step 2: Combine the fractions
Combine the fractions on the left side:
\[ \frac{a^2 + b^2}{ab} = -1 \]

### Step 3: Cross-multiply
Cross-multiply to eliminate the fraction:
\[ a^2 + b^2 = -ab \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ a^2 + b^2 + ab = 0 \]

### Step 5: Use the identity for \(a^2 + b^2\)
We can express \(a^2 + b^2\) in terms of \(ab\):
\[ a^2 + b^2 = -(ab) \]

### Step 6: Find \(\frac{a^2}{b} - \frac{b^2}{a}\)
Now, we need to find:
\[ \frac{a^2}{b} - \frac{b^2}{a} \]
This can be rewritten as:
\[ \frac{a^3 - b^3}{ab} \]

### Step 7: Factor \(a^3 - b^3\)
Using the difference of cubes formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

### Step 8: Substitute \(a^2 + b^2\)
From our earlier work, we know:
\[ a^2 + b^2 = -ab \]
Thus:
\[ a^2 + ab + b^2 = -ab + ab = 0 \]

### Step 9: Substitute back into the expression
Substituting this back into our expression:
\[ \frac{(a - b)(0)}{ab} = 0 \]

### Conclusion
Therefore, the value of \(\frac{a^2}{b} - \frac{b^2}{a}\) is:
\[ \boxed{0} \]