The points `(-a,-b)`, `(0,0)`. `(a,b)` and `(a^(2),ab))` are

Options

A. collinear
B. vertices of a rectangle
C. vertices of a parallelogram
D. None of these

Answer

To determine whether the points \((-a, -b)\), \((0, 0)\), \((a, b)\), and \((a^2, ab)\) are collinear, we can use the concept of slopes between the points. If the slopes between any two pairs of points are equal, then the points are collinear.

### Step-by-Step Solution:

1. **Identify the Points**:
Let the points be:
- \( A(-a, -b) \)
- \( B(0, 0) \)
- \( C(a, b) \)
- \( D(a^2, ab) \)

2. **Calculate the Slope of Line AB**:
The slope \( m_{AB} \) between points \( A \) and \( B \) is given by:
\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-b)}{0 - (-a)} = \frac{b}{a} \]

3. **Calculate the Slope of Line BC**:
The slope \( m_{BC} \) between points \( B \) and \( C \) is given by:
\[ m_{BC} = \frac{b - 0}{a - 0} = \frac{b}{a} \]

4. **Calculate the Slope of Line CD**:
The slope \( m_{CD} \) between points \( C \) and \( D \) is given by:
\[ m_{CD} = \frac{ab - b}{a^2 - a} = \frac{b(a - 1)}{a(a - 1)} = \frac{b}{a} \quad \text{(assuming \( a \neq 1 \))} \]

5. **Compare the Slopes**:
We have:
\[ m_{AB} = \frac{b}{a}, \quad m_{BC} = \frac{b}{a}, \quad m_{CD} = \frac{b}{a} \]
Since all slopes are equal, we conclude that the points \( A \), \( B \), \( C \), and \( D \) are collinear.

### Conclusion:
The points \((-a, -b)\), \((0, 0)\), \((a, b)\), and \((a^2, ab)\) are collinear.