The value of p for which the points (-1, 3), (2, p) and (5,-1) are collinear is 

Options

A. `-1`
B. `2`
C. `(1)/(3)`
D. `1`

Answer

To find the value of \( p \) for which the points (-1, 3), (2, p), and (5, -1) are collinear, we can use the formula for the area of a triangle formed by three points. If the area of the triangle is zero, then the points are collinear.

### Step-by-Step Solution:

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

2. **Area of Triangle Formula**: The area \( A \) of a triangle formed by the points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
For our points, this becomes:
\[ A = \frac{1}{2} \left| -1(p - (-1)) + 2(-1 - 3) + 5(3 - p) \right| \]

3. **Substituting the Values**: Substitute the coordinates into the area formula:
\[ A = \frac{1}{2} \left| -1(p + 1) + 2(-4) + 5(3 - p) \right| \]

4. **Simplifying the Expression**:
\[ A = \frac{1}{2} \left| -p - 1 - 8 + 15 - 5p \right| \]
\[ A = \frac{1}{2} \left| -6p + 6 \right| \]
\[ A = \frac{1}{2} \left| 6(1 - p) \right| \]
\[ A = 3 \left| 1 - p \right| \]

5. **Setting the Area to Zero**: For the points to be collinear, the area must be zero:
\[ 3 \left| 1 - p \right| = 0 \]

6. **Solving for \( p \)**:
\[ \left| 1 - p \right| = 0 \]
This implies:
\[ 1 - p = 0 \quad \Rightarrow \quad p = 1 \]

### Conclusion:
The value of \( p \) for which the points (-1, 3), (2, p), and (5, -1) are collinear is \( p = 1 \).