The points `(K, 3), (2, -4) and (-K+1,-2) ` are collinear. Find K. 

Answer

To find the value of \( K \) such that the points \( (K, 3) \), \( (2, -4) \), and \( (-K + 1, -2) \) are collinear, we can use the concept of the section formula and the condition for collinearity.

### Step-by-step Solution:

1. **Understanding Collinearity**:
Three points are collinear if the area of the triangle formed by them is zero. This can also be checked using the section formula.

2. **Assigning Points**:
Let the points be:
- \( A(K, 3) \)
- \( B(2, -4) \)
- \( C(-K + 1, -2) \)

3. **Using the Section Formula**:
The point \( C \) divides the line segment \( AB \) in some ratio \( m:n \). We can find this ratio by comparing the y-coordinates.

4. **Setting Up the Equation**:
The y-coordinate of point \( C \) is given by the section formula:
\[ y_C = \frac{m \cdot y_B + n \cdot y_A}{m + n} \]
Here, substituting the known values:
\[ -2 = \frac{m \cdot (-4) + n \cdot 3}{m + n} \]

5. **Assuming a Ratio**:
Let's assume the ratio is \( m:n = x:1 \). Then we can rewrite the equation as:
\[ -2 = \frac{x \cdot (-4) + 1 \cdot 3}{x + 1} \]

6. **Cross-Multiplying**:
Cross-multiplying gives:
\[ -2(x + 1) = -4x + 3 \]
Simplifying this:
\[ -2x - 2 = -4x + 3 \]

7. **Rearranging the Equation**:
Bringing all terms involving \( x \) to one side:
\[ -2 + 3 = -4x + 2x \]
This simplifies to:
\[ 1 = -2x \]

8. **Solving for \( x \)**:
Dividing both sides by -2:
\[ x = -\frac{1}{2} \]

9. **Finding the Ratio**:
The ratio \( m:n \) is \( -\frac{1}{2}:1 \) or \( 1:2 \) (after taking the absolute values).

10. **Using the x-coordinates**:
Now we can apply the section formula for the x-coordinates:
\[ x_C = \frac{m \cdot x_B + n \cdot x_A}{m + n} \]
Substituting:
\[ -K + 1 = \frac{1 \cdot 2 + 2 \cdot K}{1 + 2} \]
This simplifies to:
\[ -K + 1 = \frac{2 + 2K}{3} \]

11. **Cross-Multiplying Again**:
Cross-multiplying gives:
\[ 3(-K + 1) = 2 + 2K \]
Expanding this:
\[ -3K + 3 = 2 + 2K \]

12. **Rearranging**:
Bringing all terms involving \( K \) to one side:
\[ 3 - 2 = 2K + 3K \]
This simplifies to:
\[ 1 = 5K \]

13. **Solving for \( K \)**:
Dividing both sides by 5 gives:
\[ K = \frac{1}{5} \]

### Final Answer:
The value of \( K \) is \( \frac{1}{5} \).