For what value of k are the points (8, 1), (k, –4) and (2, –5) collinear?

Answer

To determine the value of \( k \) for which the points \( (8, 1) \), \( (k, -4) \), and \( (2, -5) \) are collinear, we can use the area of the triangle formed by these points. If the area is zero, then the points are collinear.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( A(8, 1) \), \( B(k, -4) \), and \( C(2, -5) \).

2. **Use the Area Formula**:
The area \( A \) of a triangle formed by three points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
For our points, this becomes:
\[ A = \frac{1}{2} \left| 8(-4 - (-5)) + k(-5 - 1) + 2(1 - (-4)) \right| \]

3. **Simplify the Expression**:
- Calculate \( -4 - (-5) = -4 + 5 = 1 \)
- Calculate \( -5 - 1 = -6 \)
- Calculate \( 1 - (-4) = 1 + 4 = 5 \)

Substituting these values into the area formula gives:
\[ A = \frac{1}{2} \left| 8(1) + k(-6) + 2(5) \right| \]
This simplifies to:
\[ A = \frac{1}{2} \left| 8 - 6k + 10 \right| = \frac{1}{2} \left| 18 - 6k \right| \]

4. **Set the Area to Zero**:
For the points to be collinear, the area must be zero:
\[ \frac{1}{2} \left| 18 - 6k \right| = 0 \]
This implies:
\[ \left| 18 - 6k \right| = 0 \]

5. **Solve for \( k \)**:
Since the absolute value is zero, we have:
\[ 18 - 6k = 0 \]
Rearranging gives:
\[ 6k = 18 \]
Dividing both sides by 6 results in:
\[ k = 3 \]

### Final Answer:
The value of \( k \) for which the points are collinear is \( k = 3 \).