Find the value of k if the points `A(8,1), B(k, -4) and C(2,-5)` are collinear.

Answer

To find the value of \( k \) such that the points \( A(8, 1) \), \( B(k, -4) \), and \( C(2, -5) \) 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. **Formula for Area of Triangle**:
The area \( A \) of a triangle formed by three 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 points \( A(8, 1) \), \( B(k, -4) \), and \( C(2, -5) \), we can assign:
- \( (x_1, y_1) = (8, 1) \)
- \( (x_2, y_2) = (k, -4) \)
- \( (x_3, y_3) = (2, -5) \)

2. **Substituting the Coordinates**:
Substitute the coordinates into the area formula:
\[ A = \frac{1}{2} \left| 8(-4 - (-5)) + k(-5 - 1) + 2(1 - (-4)) \right| \]
Simplifying the terms inside the absolute value:
\[ A = \frac{1}{2} \left| 8(-4 + 5) + k(-6) + 2(1 + 4) \right| \]
\[ = \frac{1}{2} \left| 8(1) - 6k + 2(5) \right| \]
\[ = \frac{1}{2} \left| 8 - 6k + 10 \right| \]
\[ = \frac{1}{2} \left| 18 - 6k \right| \]

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

4. **Solving for \( k \)**:
Rearranging the equation:
\[ 6k = 18 \]
Dividing both sides by 6:
\[ k = 3 \]

### Final Answer:
The value of \( k \) is \( 3 \).

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