One possible condition for the three points (a, b), (b, a) and `(a^(2), -b^(2))` to be collinear is

Options

A. `a - b = 2`
B. `a + b = 2`
C. `a = 1 + b`
D. `a = 1 - b`

Answer

To determine the condition for the three points \((a, b)\), \((b, a)\), and \((a^2, -b^2)\) to be collinear, we can use the concept of slopes. The points are collinear if the slope between any two pairs of points is the same.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( A = (a, b) \)
- Let \( B = (b, a) \)
- Let \( C = (a^2, -b^2) \)

2. **Calculate the Slope of Line AB**:
The slope of line \( AB \) is given by the formula:
\[ \text{slope of } AB = \frac{y_2 - y_1}{x_2 - x_1} = \frac{a - b}{b - a} = -1 \]

3. **Calculate the Slope of Line AC**:
The slope of line \( AC \) is given by:
\[ \text{slope of } AC = \frac{-b^2 - b}{a^2 - a} = \frac{-b(b + 1)}{a(a - 1)} \]

4. **Set the Slopes Equal**:
For the points to be collinear, the slopes must be equal:
\[ -1 = \frac{-b(b + 1)}{a(a - 1)} \]

5. **Cross-Multiply**:
Cross-multiplying gives:
\[ -a(a - 1) = -b(b + 1) \]
Simplifying this results in:
\[ a(a - 1) = b(b + 1) \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ a^2 - a = b^2 + b \]
This can be rewritten as:
\[ a^2 - b^2 - a - b = 0 \]

7. **Factoring**:
The left-hand side can be factored:
\[ (a - b)(a + b) - (a + b) = 0 \]
This simplifies to:
\[ (a - b)(a + b - 1) = 0 \]

8. **Conclusion**:
Thus, one possible condition for the three points to be collinear is:
\[ a - b = 0 \quad \text{or} \quad a + b - 1 = 0 \]
This means either \( a = b \) or \( a + b = 1 \).