Show that the following points are collinear :
`A(-2,1), B(-5,-1),C(1,3)`

Answer

To show that the points A(-2, 1), B(-5, -1), and C(1, 3) are collinear, we will calculate the slopes of the lines AB and AC and check if they are equal. If the slopes are equal, then the points are collinear.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( A(-2, 1) \)
- Let \( B(-5, -1) \)
- Let \( C(1, 3) \)

2. **Calculate the Slope of Line AB**:
- The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
- For points A and B:
- \( (x_1, y_1) = (-2, 1) \)
- \( (x_2, y_2) = (-5, -1) \)
- Substitute the values into the slope formula:
\[ m_{AB} = \frac{-1 - 1}{-5 - (-2)} = \frac{-2}{-5 + 2} = \frac{-2}{-3} = \frac{2}{3} \]

3. **Calculate the Slope of Line AC**:
- For points A and C:
- \( (x_1, y_1) = (-2, 1) \)
- \( (x_2, y_2) = (1, 3) \)
- Substitute the values into the slope formula:
\[ m_{AC} = \frac{3 - 1}{1 - (-2)} = \frac{2}{1 + 2} = \frac{2}{3} \]

4. **Compare the Slopes**:
- We found that:
\[ m_{AB} = \frac{2}{3} \quad \text{and} \quad m_{AC} = \frac{2}{3} \]
- Since \( m_{AB} = m_{AC} \), the points A, B, and C are collinear.

### Conclusion:
The points A(-2, 1), B(-5, -1), and C(1, 3) are collinear because the slopes of lines AB and AC are equal.