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

Options

A. The vertices of a parallelogram
B. The vertices of a rectangular
C. The vertices of a square
D. Collinear

Answer

To determine whether the points (a, b), (0, 0), (-a, -b), and (ab, b²) are collinear, we can follow these steps:

### Step 1: Identify the Points
The points given are:
1. \( P_1 = (a, b) \)
2. \( P_2 = (0, 0) \)
3. \( P_3 = (-a, -b) \)
4. \( P_4 = (ab, b^2) \)

### Step 2: Calculate the Slopes
To check if the points are collinear, we need to calculate the slopes between pairs of points. The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

#### Slope between \( P_1 \) and \( P_2 \):
Using \( P_1 = (a, b) \) and \( P_2 = (0, 0) \):
\[ m_{P_1P_2} = \frac{0 - b}{0 - a} = \frac{-b}{-a} = \frac{b}{a} \]

#### Slope between \( P_2 \) and \( P_3 \):
Using \( P_2 = (0, 0) \) and \( P_3 = (-a, -b) \):
\[ m_{P_2P_3} = \frac{-b - 0}{-a - 0} = \frac{-b}{-a} = \frac{b}{a} \]

#### Slope between \( P_3 \) and \( P_4 \):
Using \( P_3 = (-a, -b) \) and \( P_4 = (ab, b^2) \):
\[ m_{P_3P_4} = \frac{b^2 - (-b)}{ab - (-a)} = \frac{b^2 + b}{ab + a} = \frac{b(b + 1)}{a(b + 1)} \]
Assuming \( b + 1 \neq 0 \), this simplifies to:
\[ m_{P_3P_4} = \frac{b}{a} \]

#### Slope between \( P_4 \) and \( P_1 \):
Using \( P_4 = (ab, b^2) \) and \( P_1 = (a, b) \):
\[ m_{P_4P_1} = \frac{b - b^2}{a - ab} = \frac{b(1 - b)}{a(1 - b)} \]
Assuming \( 1 - b \neq 0 \), this simplifies to:
\[ m_{P_4P_1} = \frac{b}{a} \]

### Step 3: Conclusion
Since the slopes between all pairs of points are equal (\( \frac{b}{a} \)), it indicates that the points are collinear.

### Final Answer
The points (a, b), (0, 0), (-a, -b), and (ab, b²) are collinear.

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