Find the value of 'k', for which the given points are colinear.
(8,1), (k,-4), (2,-5)

Answer

To find the value of 'k' for which the points (8, 1), (k, -4), and (2, -5) are collinear, we can use the concept that the area of the triangle formed by these three points must be zero. This is because if the points are collinear, they do not form a triangle.

### Step-by-Step Solution:

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

2. **Use the Area Formula**:
The area \( A \) of the triangle formed by the points (x1, y1), (x2, y2), and (x3, y3) is given by 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, substituting in the coordinates:
\[ A = \frac{1}{2} \left| 8(-4 + 5) + k(-5 - 1) + 2(1 + 4) \right| \]

3. **Simplify the Expression**:
This simplifies to:
\[ A = \frac{1}{2} \left| 8(1) + k(-6) + 2(5) \right| \]
\[ = \frac{1}{2} \left| 8 - 6k + 10 \right| \]
\[ = \frac{1}{2} \left| 18 - 6k \right| \]

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

5. **Solve the Absolute Value Equation**:
Therefore, we have:
\[ 18 - 6k = 0 \]
Rearranging gives:
\[ 6k = 18 \]
Dividing both sides by 6:
\[ k = 3 \]

### Conclusion:
The value of \( k \) for which the points (8, 1), (k, -4), and (2, -5) are collinear is \( k = 3 \).

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