If the roots of the equations `x^(2) - (a-b)x + c = 0` are integers which are equal in magnitude and opposite in sign, which of the following statements are true ?
I. `a = - b`
II. `a =b`
III.`c+a^(2) = 0`

Options

A. Only I
B. Only II
C. Only III
D. Both I and II

Answer

To solve the problem, we need to analyze the given quadratic equation \(x^2 - (a-b)x + c = 0\) and the conditions on its roots.

### Step 1: Identify the roots
Since the roots are integers that are equal in magnitude and opposite in sign, we can denote the roots as \(\alpha\) and \(-\alpha\).

### Step 2: Use the sum of the roots
From Vieta's formulas, we know that the sum of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is given by \(-\frac{b}{a}\). In our case, the sum of the roots \(\alpha + (-\alpha) = 0\) must equal the expression for the sum of the roots from the equation:

\[ \alpha + (-\alpha) = -(a - b) \implies 0 = -(a - b) \]

This simplifies to:

\[ a - b = 0 \implies a = b \]

### Step 3: Use the product of the roots
Next, we consider the product of the roots, which is given by \(\alpha \cdot (-\alpha) = -\alpha^2\). According to Vieta's formulas, the product of the roots is also given by \(\frac{c}{1}\) (since the coefficient of \(x^2\) is 1):

\[ \alpha \cdot (-\alpha) = c \implies -\alpha^2 = c \]

This can be rearranged to:

\[ c + \alpha^2 = 0 \implies c = -\alpha^2 \]

### Step 4: Analyze the statements
Now we can evaluate the given statements:

1. **Statement I: \(a = -b\)**
From our earlier conclusion, we found that \(a = b\). Therefore, this statement is **false**.

2. **Statement II: \(a = b\)**
We derived that \(a = b\) from the sum of the roots. Therefore, this statement is **true**.

3. **Statement III: \(c + a^2 = 0\)**
We found that \(c = -\alpha^2\). Since \(a = b\), we can express \(c + a^2\) as follows:
\[ c + a^2 = -\alpha^2 + a^2 = 0 \quad \text{(if } a^2 = \alpha^2\text{)} \]
Thus, this statement is also **true**.

### Conclusion
The true statements are:
- II: \(a = b\)
- III: \(c + a^2 = 0\)

Thus, the correct answer is that statements II and III are true.